Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

Jeffrey Brock 1 , Christopher Leininger 2 , Babak Modami 3  and Kasra Rafi 4
  • 1 Department of Mathematics, RI, Providence, USA
  • 2 Department of Mathematics, 1409 W Green ST, Urbana, USA
  • 3 Department of Mathematics, 1409 W Green ST, Urbana, USA
  • 4 Department of Mathematics, ON, Toronto, Canada
Jeffrey Brock, Christopher Leininger
  • Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green ST, Urbana, IL, USA
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, Babak Modami
  • Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green ST, Urbana, IL, USA
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and Kasra Rafi

Abstract

Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension 3g-3, for example.

We also study the limit sets in the Thurston boundary of Teichmüller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the 1-skeleton of the simplex of projective classes of measures visiting every vertex.

1 Introduction

This paper builds on the work of the second and fourth author with Anna Lenzhen, [22], in which the authors construct a sequence of curves in the five-punctured sphere S with the following properties (see Section 2 for definitions). First, the sequence is a quasi-geodesic ray in the curve complex of S, and hence converges to some ending lamination ν. Second, ν is nonuniquely ergodic, and the sequence naturally splits into two subsequences, each of which converges to one of the ergodic measures on ν in the space of projective measured laminations. Third, for any choice of measure ν¯ on ν and base point X in Teichmüller space, the Teichmüller ray based at X and defined by the quadratic differential with vertical foliation ν¯, accumulates on the entire simplex of measures on ν in the Thurston compactification. The construction in [22] was actually a family of sequences depending on certain parameters.

In this paper we extract the key features of the sequences produced in the above construction as a set of local properties for any sequence of curves {γk}k=0 on any surface, which we denote 𝒫; see Section 3 and Definition 3.1 as well as Section 7 for examples. Here, “local” is more precisely m-local for some 2mξ(S) (where ξ(S)=dim(Teich(S))), and means that the conditions in 𝒫 involve relations between curves contained subsets of the form {γk,,γk+2m} for k0. We refer to the number m as the subsequence counter. Most of the conditions in 𝒫 are stated in terms of intersection numbers, though they also include information about twisting which is recorded in an auxiliary sequence {ek}k=0.

Theorem 1.1.

For appropriate choices of parameters in P, any sequence {γk}k=0 in C(S) satisfying P will be the vertices of a quasi-geodesic in C(S) and hence will limit to an ending lamination ν in C(S)EL(S).

If μ=γ0γm-1, then for any km, the subsequence counter, we have

dγk(μ,ν)+ek.

On the other hand, there is a constant R>0 with the property that for any proper subsurface Wγk for any kN we have

dW(μ,ν)<R.

See Propositions 4.4 and 4.5 for precise statements. Here dW is the projection coefficient for W and dγ the projection coefficient for (the annular neighborhood of) γ; see Section 2.4.

Although the conditions in 𝒫 only provide local information about intersection numbers, we can deduce estimates on intersection numbers between any two curves in the sequence from this; see Theorem 5.1. From these estimates, we are able to promote the convergence γkν in 𝒞(S)¯ into precise information about convergence in 𝒫(S). To state this, we note that the local condition depends on the subsequence counter m. There are m subsequences {γih}i=0, for h=0,,m-1, defined by γih=γim+h.

Theorem 1.2.

For appropriate choices of parameters in P, and any sequence {γk}k=0 in C(S) satisfying P, the ending lamination νEL(S) from Theorem 1.1 is nonuniquely ergodic. Moreover, if m is the subsequence counter, then the dimension of the space of measures on ν is precisely m, and the m subsequences {γih}i=0 converge to m ergodic measures ν¯h on ν, for h=0,,m-1, spanning the space of measures.

For precise statements, see Theorems 5.10, 6.1, and 6.5.

We note that for any nonuniquely ergodic lamination ν, the space of measures is always the cone on the simplex of measures on ν, denoted Δ(ν), which is projectively well-defined. The vertices of Δ(ν) are the ergodic measures, and the dimension of the space of measures is at most ξ(S): This follows from the fact that the Thurston symplectic form on the 2ξ(S)-dimensional space (S) must restrict to zero on the cone on Δ(ν) since it is bounded above by the geometric intersection number, [35, Section 3.2], and consequently must be at most half-dimensional (see also [28, Section 1] and the reference to [40, 20]). We note that the subsequence counter m can also be at most ξ(S), and the explicit constructions in Section 7 are quite flexible and provide examples with this maximal dimension, as well as examples with smaller dimensions.

As an application of these theorems, together with the main result of the first and third author in [5] and Theorem 1.1, we have:

Corollary 1.3.

Suppose that ν is as in Theorem 1.1. Any Weil–Petersson geodesic ray with forward ending lamination ν is recurrent to a compact subset of the moduli space.

Here, the ending lamination of a Weil–Petersson geodesic ray is given as in [3, 4]. The corollary, which follows directly from [5, Theorem 4.1] after observing that ν satisfies the condition of nonannular bounded combinatorics (see Proposition 4.5), provides greater insight into the class of Weil–Petersson ending laminations that violate Masur’s criterion. In particular, these nonuniquely ergodic laminations determine recurrent Weil–Petersson geodesic rays, by contrast to the setting of Teichmüller geodesics where Masur’s criterion [30] guarantees a Teichmüller geodesic with such a vertical foliation diverges.

For any lamination ν coming from a sequence {γk}k=0 satisfying 𝒫, as well as some additional conditions (see (8.8) in Section 8 and condition $\mathcal{P}$ (iv) in Section 9), we analyze the limit set of a Teichmüller geodesic ray defined by a quadratic differential with vertical foliation ν¯ supported on ν. To describe our result about the limiting behavior of this geodesic ray, we denote the simplex of the projective classes of measures supported on the lamination by Δ(ν) in the space of projective measured foliations, viewed as the Thurston boundary of Teichmüller space.

Theorem 1.4.

Suppose that ν is the limiting lamination of a sequence {γk}k=0 satisfying the conditions P, $\mathcal{P}$ (iv), and (8.8). Let

ν¯=h=0m-1xhν¯h,

where xh>0 for h=0,,m-1, and let r:[0,)Teich(S) be a Teichmüller geodesic ray with vertical measured lamination ν¯. Then the limit set of r in the Thurston boundary is the simple closed curve in the simplex Δ(ν) of measures on ν that is the concatenation of edges

[[ν¯0],[ν¯1]][[ν¯1,ν¯2]][[ν¯m-1],[ν¯0]].

When m3, the theorem shows that there are Teichmüller geodesics whose limit set does not contain any point in the interior of Δ(ν). In addition, it answers the following question raised by Jonathan Chaika.

Question 1.5.

Is the limit set of each Teichmüller geodesic ray simply connected?

For m3, the theorem shows that answer to this question is no. Namely, Teichmüller geodesic rays with vertical measured lamination as above provide examples of geodesics with limit set being a topological circle, and hence not simply connected.

The results of this paper (as well as those of [22]) were inspired by work of Masur in [29], Lenzhen [23], and Gabai [14]. In [29] Masur showed that if ν is a uniquely ergodic foliation, then any Teichmüller ray defined by a quadratic differential with vertical foliation supported on ν limits to [ν] in the Thurston compactification. Lenzhen [23] gave the first examples of Teichmüller rays which do not converge in the Thurston compactification. Lenzhen’s rays were defined by quadratic differentials with non-minimal vertical foliations, and in both [22] and [8], nonconvergent rays defined by quadratic differentials with minimal vertical foliations were constructed. The methods in these two papers are quite different, and as mentioned above, the approach taken in this paper is more closely related to that of [22]. We also note the results of this paper, as well as [23, 22, 8], are in sharp contrast to the work of Hackobyan and Saric in [17] where it is shown that Teichmüller rays in the universal Teichmüller space always converge in the corresponding Thurston compactification.

Our example of nonuniquely ergodic laminations obtained from a sequence of curves are similar to those produced by Gabai in [14]. On the other hand, our construction provides additional information, especially important are the estimates on intersection numbers and subsurface projections, that allow us to study the limiting behavior of the associated Teichmüller rays. For more on the history and results about the existence and constructions of nonuniquely ergodic laminations and the study of limit sets of Teichmüller geodesics with such vertical laminations we refer the reader to the introduction of [22].

2 Background

We use the following notation throughout this paper.

Notation 2.1.

Let K1 and C0 and let f,g:X be two functions. We write

  1. f+Cg if f(x)-Cg(x)f(x)+C for all xX,
  2. f*Kg if 1Kf(x)g(x)Kf(x) for all xX,
  3. fK,Cg if 1K(f(x)-C)g(x)Kf(x)+C for all xX,
  4. f*Kg if f(x)Kg(x) for all xX,
  5. f+Cg if f(x)g(x)+C for all xX,
  6. fK,Cg if f(x)Kg(x)+C for all xX.

When the constants are known from the text we drop them from the notations. Finally, we also write f=O(g) if f*g.

Let S=Sg,b be an orientable surface of finite type with genus g and b holes (a hole can be either a puncture or a boundary component). Define the complexity of S by ξ(S)=3g-3+b. The main surface we will consider will have ξ>1 and all holes will be punctures. However, we will also be interested in subsurfaces and covers of the main surface, which can also have ξ1. For surfaces S with ξ(S)1, we will equip it with a reference metric, which is any complete, hyperbolic metric of finite area with geodesic boundary (if any).

2.1 Curve complexes

For any surface Y, ξ(Y)1, the curve complex of Y, denoted by 𝒞(Y), is a flag complex whose vertices are the isotopy classes of simple closed curves on Y that are essential, meaning non-null homotopic and nonperipheral. For ξ(Y)>1, a set of k+1 distinct isotopy classes of curves defines a k-simplex if any pair can be represented by disjoint curves. For ξ(Y)=1 (Y is S0,4 or S1,1), the definition is modified as follows: a set of k+1 distinct isotopy classes defines a k-simplex if the curves can be represented intersecting twice (for Y=S0,2) or once (for Y=S1,1).

The only surface Y with ξ(Y)<1 of interest for us is a compact annulus with two boundary components. These arise as follows. For any essential simple closed curve α on our main surface S, let Yα denote the annular cover of S to which α lifts. The reference hyperbolic metric on S lifts and provides a compactification of this cover by a compact annulus with boundary (which is independent of the metric). The curve complex of α, denoted 𝒞(Yα), or simply 𝒞(α), has vertex set being the properly embedded, essential arcs in Yα, up to isotopy fixing the boundary pointwise. A set of isotopy classes of arcs spans a simplex if any pair can be realized with disjoint interiors.

Distances between vertices in 𝒞(Y) (for any Y) will be measured in the 1-skeleton, so the higher-dimensional simplices are mostly irrelevant. Masur and Minsky [31] proved that for any Y, there is a δ>0 so that 𝒞(Y) is δ-hyperbolic.

For surfaces Y with ξ(Y)1, we also consider the arc and curve complex 𝒜𝒞(Y), defined in a similar way to 𝒞(Y). Here vertices are isotopy classes of essential simple closed curves and essential, properly embedded arcs (isotopies need not fix the boundary pointwise), with simplices defined again in terms of disjoint representatives. Arc and curve complexes are quasi-isometric to curve complexes, and so are also δ-hyperbolic.

Multicurves (respectively, multiarcs) are disjoint unions of pairwise nonisotopic essential simple closed curves (respectively, simple closed curves and properly embedded arcs). Up to isotopy a multicurve (respectively, multiarc) determines, and is determined by, a simplex in 𝒞(S) (respectively, 𝒜𝒞(S)). A marking μ is a pants decomposition base(μ), called the base of μ, together with a transversal curve βα, for each αbase(μ), which is a curve minimally intersecting α and disjoint from base(μ)-α. A partial marking μ is similarly defined, but not every curve in the pants decomposition base(μ) is required to have a transversal curve.

For more details on curve complexes, arc and curve complexes, and markings, we refer the reader to [31].

Remark 2.2.

When the number ξ(S) is at least 1, it is equal to the number of curves in a pants decomposition. When all the holes of S are punctures, ξ(S) is also the complex dimension of Teichmüller space of S.

2.2 Laminations and foliations

A lamination will mean a geodesic lamination (with respect to the reference metric if no other metric is specified), and a measured lamination is a geodesic lamination ν, called the support, with an invariant transverse measure ν¯. We will often refer to a measured lamination just by the measure ν¯ (as this determines the support ν). The space of all measured laminations will be denoted (S), and for any two metrics, the resulting spaces of measured laminations are canonically identified. By taking geodesic representatives, simple closed curves and multicurves determine geodesic laminations. Weighted simple closed curves and multicurves determine measured laminations are dense in (S), and the geometric intersection number extends to a continuous, bi-homogeneous function

i:(S)×(S).

By a measured foliation on S we will mean a singular measured foliation with prong singularities of negative index (and at punctures, filling in the puncture produces a k-prong singularity with k1). When convenient, a measured foliation may be considered only defined up to measure equivalence, and the space of measure equivalence classes of measured foliations is denoted (S). The spaces (S) and (S) are canonically identified, and we will frequently not distinguish between measured laminations and measured foliations. A foliation or lamination is uniquely ergodic if it supports a unique (up to scaling) transverse measure, or equivalently, if the first return map to (the double of) any transversal is uniquely ergodic. Otherwise it is nonuniquely ergodic. We write 𝒫(S) and 𝒫(S) for the quotient spaces, identifying measured laminations or foliations that differ by scaling the measure. See [35, 7, 13, 39, 25] for complete definitions, detailed discussion, and equivalence of (S) and (S).

2.3 Gromov boundary of the curve complex

A lamination ν on S is called an ending lamination if it is minimal (every leaf is dense) and filling (every simple closed geodesic on the surface nontrivially, transversely intersect ν). Every ending lamination admits a transverse measure, and we let (S) denote the space of all ending laminations. This is the quotient space of the subspace of (S) consisting of measured laminations supported on ending laminations, by the map which forgets the measures. The following theorem of Klarreich [21] identifies the Gromov boundary of 𝒞(S) with (S).

Theorem 2.3 (Boundary of the curve complex).

There is a homeomorphism Φ from the Gromov boundary of C(S) equipped with its standard topology to EL(S).

Let {γk}k=0 be a sequence of curves in C0(S) that converges to a point x in the Gromov boundary of C(S). Regarding each γk as a projective measured lamination, any accumulation point of the sequence {γk}k=0 in PML(S) is supported on Φ(x).

We will use this theorem throughout to identify points in 𝒞(S) with ending laminations in (S).

2.4 Subsurface coefficients

An essential subsurfaceY of a surface Z with ξ(Y)1 is a closed, connected, embedded subsurface whose boundary components are either essential curves in Z or boundary component of Z, and whose punctures are punctures of Z. All such subsurfaces are considered up to isotopy, and we often choose representatives that are components of complements of small neighborhoods of simple closed geodesics, which therefore have minimal, transverse intersection with any lamination. The only essential subsurfaces Y of Z with ξ(Y)<1 are not actually subsurfaces at all, but rather such a Y is the compactified annular covers Yα of Z associated to a simple closed curve α in Z. We sometimes confuse an annular neighborhood of α with the cover Yα (hence the reference to it as a subsurface) when convenient. We will always write YZ to denote an essential subsurface, even when it is not, strictly speaking, a subset of Z.

Let YZ be an essential nonannular subsurface and λ a lamination (possibly a multicurve) and we define the subsurface projection of λ to Y. Represent Y as a component of the complement of a very small neighborhood of geodesic multicurve. If λY=, then define πY(λ)=. Otherwise, πY(λ) is the union of all curves which are (i) simple closed curve components of Yλ or (ii) essential components of N(aY), where aλY is any arc, and N(aY) is a regular neighborhood of the union. If Yα is an essential annular subsurface, then πYα(λ), or simply πα(λ), is defined as follows. For any component of the preimage of λ in the annular cover corresponding to α, the closure is an arc in Yα, and we take the union of all such arcs that are essential (that is, the arcs that connect the two boundary components).

For a marking μ (or a partial marking), if Y=Yα is an annulus with core curve αbase(μ), then πY(μ)=πα(βα), where βα is the transverse curve for α in μ. Otherwise, πY(μ)=πY(base(μ)). For any lamination or partial marking λ and any essential subsurface Y, πY(λ) is a subset of diameter at most 2.

Let μ,μ be laminations, multiarcs, or partial markings on Z and YZ an essential subsurface. The Y-subsurface coefficient of μ and μ is defined by

dY(μ,μ):=diam𝒞(Y)(πY(μ)πY(μ)).

Remark 2.4.

The subsurface coefficient is sometimes alternatively defined as the (minimal) distance between πY(μ) and πY(μ). Since the diameter of the projection of any marking or lamination is bounded by 2, these definitions differ by at most 4. The definition we have chosen satisfies a triangle inequality (when the projections involved are nonempty), which is particular useful for our purposes.

The following lemma provides an upper bound for a subsurface coefficient in terms of intersection numbers.

Lemma 2.5 ([32, Section 2]).

Given curves α,βC(S), for any essential subsurface YS we have

dY(α,β)2i(α,β)+1.

When Y is an annular subsurface, the above bound holds with multiplicative factor 1.

Remark 2.6.

The bound in the above lemma can be improved to logi(α,β) for ξ(Y)1, but the bound given is sufficient for our purposes.

The following result equivalent to [9, Corollary D] provides for a comparison between the logarithm of intersection number and sum of subsurfaces coefficients. For a pair of markings μ,μ, the intersection number i(μ,μ) is defined to be the sum of the intersection numbers of the curves in μ with those in μ.

Theorem 2.7.

Given A>0 sufficiently large, there are constants so that for any two multi-curves, multi-arcs or markings μ and μ we have

logi(μ,μ)WY𝑛𝑜𝑛𝑎𝑛𝑛𝑢𝑙𝑎𝑟{dW(μ,μ)}A+WY𝑎𝑛𝑛𝑢𝑙𝑎𝑟log{dW(μ,μ)}A,

where W is so that μ,μW.

In this theorem, {}A is a cut-off function defined by {x}A=x if xA, and {x}A=0 if x<A.

Notation 2.8.

Given a lamination or a partial marking μ and subsurface Y, we say that μ and Y overlap, writing μY if πY(μ). For any marking μ and any subsurface Y, we have μY. Given two subsurfaces Y and Z, if YZ and ZY, then we say that Y and Z overlap, and write YZ.

The inequality first proved by J. Behrstock [1] relates subsurface coefficients for overlapping subsurfaces.

Theorem 2.9 (Behrstock inequality).

There is a constant B0>0 so that given a partial marking or lamination μ and subsurfaces Y and Z satisfying YZ we have

min{dY(Z,μ),dZ(Y,μ)}B0

whenever μY and μZ.

Remark 2.10.

As shown in [26], the constant B0 can be taken to be 10. In fact, if one projection is at least 10, then the other is 4.

The following theorem is a straightforward consequence of the Bounded Geodesic Image Theorem [32, Theorem 3.1].

Theorem 2.11 (Bounded geodesic image).

Given k1 and c0, there is a constant G>0 with the following property. Let YS be a subsurface. Let {γk}i=0 be a 1-Lipschitz (k,c)-quasi-geodesic in C(S) so that γkY for all i. Then diamY({γk}i=0)G.

2.5 Teichmüller theory

Throughout the paper, we assume that S is a surface and that any holes of S are punctures. The Teichmüller space of S, Teich(S), is the space of equivalence classes of marked complex structures [f:SX], where f is an orientation preserving homeomorphism to a finite-type Riemann surface X, where (f:SX)(g:SY) if fg-1 is isotopic to a conformal map. We often abuse notation, and simply refer to X as a point in Teichmüller space, with the equivalence class of marking implicit. We equip Teich(S) with the Teichmüller metric, whose geodesics are defined in terms of quadratic differentials.

Let X be a finite-type Riemann surface and let T(1,0)*X be the holomorphic cotangent bundle of X. A quadratic differential q is a nonzero, integrable, holomorphic section of the bundle T(1,0)*XT(1,0)*X. In local coordinates q has the form q(z)dz2, where q(z) is holomorphic function. Changing to a different coordinate w, q changes by the square of the derivative, and is thus given by q(z(w))(wz)2dw2. The integrability condition is only relevant when X has punctures, in which case it guarantees that q has at worst simple poles at the punctures.

In local coordinates away from zeros of q the quadratic differential q determines the 1-form q(z)dz2. Integrating this 1-form determines a natural coordinateζ=ξ+iη. Then the trajectories of dξ0 and dη0, respectively, determine the horizontal and vertical foliations of q on X. Integrating |dξ| and |dη| determines transverse measures on vertical and horizontal foliations, respectively. These extend to measured foliations on the entire surface S with singularities at the zeros. Using the identification (S)(S), we often refer to the vertical and horizontal measured laminations of q.

Now given a quadratic differential q on X, the associated Teichmüller geodesic is determined by the family of Riemann surfaces Xt defined by local coordinates ζt=etξ+e-tη, where ζ=ξ+iη is a natural coordinate of q at X and t. Every Teichmüller geodesic ray based at X is determined by a quadratic differential q on X. See [15] for details on Teichmüller space and quadratic differentials.

2.6 The Thurston compactification

Given a point [f:SX] in Teich(S) and a curve α, the hyperbolic length of α at [f:SX] is defined to be hyperbolic length of the geodesic homotopic to f(α) in X. Again abusing notation and denoting the point in Teich(S) by X, we write the hyperbolic length simply as HypX(α). The hyperbolic length function extends to a continuous function

Hyp()():Teich(S)×(S).

The Thurston compactification, Teich(S)¯=Teich(S)𝒫(S), is constructed so that a sequence {Xn}Teich(S) converges to [ν¯]𝒫(S) if and only if

limnHypXn(α)HypXn(β)=i(ν¯,α)i(ν¯,β)

for all simple closed curves α,β with i(ν¯,β)0. See [2, 13] for more details on the Thurston compactification.

2.7 Some hyperbolic geometry

Here we list a few important hyperbolic geometry estimates. For a hyperbolic metric XTeich(S) and a simple closed curve α, in addition to the length HypX(α), we also have the quantity wX(α), the width of α in X. This is the width of a maximal embedded tubular neighborhood of α in the hyperbolic metric X – that is, wX(α) is the maximal w so that the open w/2-neighborhood of α is an annular neighborhood of α. The Collar Lemma (see e.g. [6, Section 4.1]) provides a lower bound on the width:

Lemma 2.12.

For any simple closed curve α, we have

wX(α)2sinh-1(1sinh(HypX(α)/2)).

Consequently,

wX(α)+2log(1HypX(α)).

The second statement comes from the first, together with an easy area argument. The implicit additive error depends only on the topology of S.

We also let ϵ0>0 be the Margulis constant, which has the property that any two hyperbolic geodesics of length at most ϵ0 must be embedded and disjoint.

2.8 Short markings

For L>0 sufficiently large, an L-bounded length marking at XTeich(S) (or L-short marking) is a marking with the property that any curve in base(μ) has hyperbolic length less than L, and so that for each αbase(μ), the transversal curve to α has smallest possible length in X. Choosing ϵ sufficiently large (larger than the Bers constant of the surface) the distance between any two points in Teichmüller space can be estimated up to additive and multiplicative error in terms of the subsurface coefficients of the short markings at those points, together with the lengths of their base curves; see [37].

3 Sequences of curves

Over the course of the next three sections we will provide general conditions on a sequence of curves which guarantee that any accumulation point in 𝒫(S) of this sequence is a nonuniquely ergodic ending lamination. In [14, Section 9], Gabai describes a construction of minimal filling nonuniquely ergodic geodesic laminations. The construction is topological in nature. Our construction in this paper and that of [22] can be considered as quantifications of Gabai’s construction where the estimates for intersection numbers are computed explicitly. These estimates allow us to provide more detailed information about the limits in 𝒫(S) as well as limiting behavior of associated Teichmüller geodesics.

In this section we state conditions a sequence of curves can satisfy, starting with an example, and describe a useful way of mentally organizing them. The conditions are motivated by the examples in [22], and so we recall that construction to provide the reader concrete examples to keep in mind. A more robust construction that illustrates more general phenomena is detailed in Section 7.

Throughout the rest of this paper {ek}k=0 is an increasing sequence of integers satisfying

ek+1aekfor any k0,

where a>1. Consequently, for all l<k, we have ekak-lel.

3.1 Motivating example

The motivating examples are sequences of curves in S0,5, the five-punctured sphere. We view this surface as the double of a pentagon minus its vertices over its boundary. This description provides an obvious order five rotational symmetry ρ obtained by rotating the pentagon counter-clockwise by an angle 4π/5. Let γ0 be a curve which is the boundary of a small neighborhood of one of the sides of the pentagon and let γ=ρ2(γ0) (see Figure 1). Write 𝒟=𝒟γ for the positive Dehn twist about γ.

Now define γk to be the image of γ0 under a composition of powers of 𝒟 and ρ by the following formula:

γk=𝒟e2ρ𝒟e3ρ𝒟ekρ𝒟ek+1ρ(γ0).

The first five curves, γ0,,γ4, in the sequence are shown in Figure 1.

Figure 1
Figure 1

The curves γ0,,γ4 in S0,4. Any five consecutive curves γk-2,,γk+2 differ from those shown here by a homeomorphism, and replacing e2 by ek.

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0024

Observe that for any k3, the four consecutive curves γk-2,,γk+1 are just the image of γ0,,γ3 under the homeomorphism

Φk-1=𝒟e2ρ𝒟ek-1ρ.

Furthermore, the next curve in the sequence, γk+2, is the image of 𝒟ekρ(γ3). In particular, up to homeomorphism, any five consecutive curves γk-2,,γk+2 in the sequence appear as in Figure 1 with e2 replaced by ek.

3.2 Intersection conditions

We now describe the general conditions, and verify that the above sequence of curves satisfies them. To begin, we fix positive integers b1bb2. We will also assume that e0>E+G (and hence by (3.1) ek>ak(E+G) for all k), where G is the constant from Theorem 2.11 and E is the constant in Theorem 4.1 below. For the examples in S0,5 described above, we will have b=b1=b2=2.

In the next definition, 𝒟γ is the Dehn twist in a curve γ.

Definition 3.1.

Suppose that mξ(S), and assume that b,b1,b2,a, and {ek}k=0 are as above. We say that a sequence of curves {γk}k=0 on S satisfies 𝒫 if the following properties hold for all k0:

  1. (i)γk,,γk+m-1 are pair-wise disjoint and distinct,
  2. (ii)γk,,γk+2m-1 fill the surface S,
  3. (iii)γk+m=𝒟γkek(γk+m), where γk+m is a curve such that
    i(γk+m,γj){[b1,b2]for j{k-m,,k-1},=bfor j=k,=0for j{k+1,,k+m-1},
    (here we ignore any situation with j<0).

We will wish to impose some additional constraints on the constant a (specifically, we will require it to be chosen so that (5.4) holds), and so in the notation we sometimes express the dependence on a writing 𝒫=𝒫(a). Of course, 𝒫 depends on the choice of constants b1bb2 and the sequence {ek}, but we will impose no further constraints on the b constants, and the conditions on {ek} depend on a.

Here we verify that the sequence of curves on S0,5 described above satisfies these conditions with m=2. Note that the conditions are all “local”, meaning that they involve a consecutive sequence of at most 2m+1 curves – for our example, that is a sequence of at most five consecutive curves. As noted above, any five consecutive curves γk-2,,γk+2 differ from those in Figure 1 by applying the homeomorphism Φk-1=𝒟e2ρ𝒟ek-1ρ, and changing e2 to ek. From this, it is straight forward to verify that this sequence satisfies these conditions.

Since m=2, condition (i) says that two consecutive curves are disjoint, while condition (ii) says that four consecutive curves fill S0,5. Note that (i) is clearly true for γ0,γ1 and (ii) for γ0,,γ3. Since any two or four consecutive curves differ from these by a homeomorphism, conditions (i) and (ii) hold for all k.

Finally, note that γ4=𝒟γe2ρ(γ3), and so setting γ4=ρ(γ3) and observing that γ=γ2, (iii) clearly holds for k=2 by inspection of Figure 1. The case for general k follows from this figure as well, after applying Φk-1. Specifically, γk+2 is obtained from ρ(γ3) by applying Φk-1𝒟γ2ek, or equivalently, setting γk+2=Φk-1(ρ(γ3)),

γk+1=Φk-1𝒟γ2ekΦk-1-1(Φk-1(ρ(γ3)))=𝒟Φk-1(γ2)ek(γk+2)=𝒟γkek(γk+2).

Since γk-2,γk-1,γk,γk+1,γk+2,γk+2 are the images of γ0,γ1,γ2,γ3,γ4,γ4, respectively, under Φk-1, condition (iii) follows for general k by inspection of Figure 1.

Returning to the general case, we elaborate a bit on the properties in 𝒫. First we make a simple observation.

Lemma 3.2.

For every j,k0 with j{k-m+1,,k}, we have

i(γk+m,γj)[b1,b2],

moreover i(γk+m,γk)=b.

Proof.

Since γk+m=𝒟γkek(γk+m) and 𝒟γk(γj)=γj (because i(γj,γk)=0), it follows that

i(γk+m,γj)=i(𝒟γk-ek(γk+m),𝒟γk-ek(γj))=i(γk+m,γj)[b1,b2]

proving the first statement. For the special case j=k, i(γk+m,γk)=b, and the second statement follows. ∎

3.3 Visualizing the conditions of 𝒫

The conditions imposed in 𝒫 involve intervals of length m and 2m, as well as mod m congruence conditions. It is useful to view the tail of the sequence starting at any curve γi (for example, when i=0 this is the entire sequence), in the following form:

article image

From the first condition of 𝒫, all curves in any row are pairwise disjoint. Lemma 3.2 tells us that γi intersects the curve directly below it b times and it intersects everything in the row directly below it between b1 and b2 times. The second condition in 𝒫 tells us that any two consecutive rows fill S. The third condition (part of which is used in the proof of Lemma 3.2), can be thought of as saying that going straight down two rows from γi to γi+2m gives a curve that “almost” differs by the power of the Dehn twist 𝒟γi+mei+m. To understand this interpretation, note that γi+2m and γi+2m differ precisely by this power of a twist, while on the other hand, each of γi+2m and γi have intersection number at most b2 with the filling set γi,,γi+2m-1 (which we view as saying that γi and γi+2m are “similar”).

4 Curve complex quasi-geodesics

The purpose of this section is to provide general conditions (Theorem 4.1) on a sequence of subsurfaces in terms of subsurface coefficients of consecutive elements which guarantee that their boundaries define a quasi-geodesic in the curve complex of the surface. Appealing to Theorem 2.3, we deduce that such sequences determine an ending lamination. We end by proving that a sequence of curves satisfying 𝒫 are core curves of annuli satisfying the conditions of Theorem 4.1, and hence are vertices of a quasi-geodesic in 𝒞(S) defining an ending lamination ν(S).

Variations of this result appeared in [26], [11], [34], [22], and [5] for example. Here our conditions only involve the intersection pattern and projection coefficients of fixed number of consecutive subsurfaces along the sequence. In this sense these are local conditions.

Theorem 4.1 (Local to global).

Given a surface S and 2mξ(S), there are constants E>C>0 with the following properties. Let {Yk}k=0 be a sequence of subsurfaces of S. Suppose that for each integer k0,

  1. (1)the multi-curves Yk,,Yk+m-1 are pairwise disjoint,
  2. (2)YkYj for all j{k+m,,k+2m-1},
  3. (3)dYk(Yj,Yj)>E for any j{k+m,,k+2m-1}, j{k-2m+1,,k-m}.

Then for every j,j,k with jk+m and jk-m we have

YkYj𝑎𝑛𝑑YkYj

and

dYk(Yj,Yj)dYk(Yk-m,Yk+m)-C.

Furthermore, suppose that for some n1 and all k0,

  1. (4)the multi-curves Yk,,Yk+2n-1 fill S.

Then for any two indices k,j0 with |k-j|2n we have

dS(Yj,Yk)|k-j|4n-(m2n+1).

In the hypotheses (as well as the conclusions) of this theorem, we ignore any condition in which there is a negative index.

Proof.

Set the constants

C=2B0+4+G and E=C+B0+G+4.

Here B0 is the constant from Theorem 2.9 (Behrstock inequality) and G is the constant from Theorem 2.11 (Bounded geodesic image theorem) for a geodesic (i.e. k=1, c=0). We prove (4.1) and (4.2) simultaneously by a double induction on (j-k,k-j).

For the base of induction, suppose that mk-j2m-1 and mj-k2m-1. Statement (4.1) follows from (2). To prove (4.2) note that by (1)Yk+m,,Yj are pairwise disjoint and have non-empty projections to Yk. Consequently, the distance in Yk between any two of these boundaries is at most 2, and so

diamYk({Yl}l=k+mj)2.

Similarly, diamYk({Yl}l=jk-m)2. By the triangle inequality we have

dYk(Yj,Yj)dYk(Yk-m,Yk+m)-dYk(Yj,Yk+m)-dYi(Yk-m,Yj)
dYk(Yk-m,Yk+m)-4dYk(Yk-m,Yk+m)-C,

which is the bound (4.2).

Suppose that (4.1) and (4.2) hold for all mk-j2m-1 and mj-kN, for some N2m-1. We suppose j-k=N+1 and we must prove both (4.1) and (4.2) for (j-k,k-j).

From the base of induction we already have YkYj. To complete the proof of (4.1), we prove YkYj. Since m=(k+m)-k2m-1 and mj-(k+m)=N+1-mN, from the inductive hypothesis we have

YkYk+mandYjYk+m

and

dYk+m(Yk,Yj)dYk+m(Yk,Yk+2m)-CE-C4.

Consequently, i(Yk,Yj)0 and YkYj as required.

We now turn to the proof of (4.2). Since YkYj and YkYj, by (2) we may write the following triangle inequality:

dYk(Yj,Yj)dYk(Yk-m,Yk+m)-dYk(Yk-m,Yj)-dYk(Yj,Yk+m).

Since mj-(k+m)=N+1-mN, from the inductive hypothesis we have

dYk+m(Yk,Yj)dYk+m(Yk,Yk+2m)-CE-CB0.

By Theorem 2.9, dYk(Yk+m,Yj)B0. On the other hand, as in the proof of the base case of induction, since mk-j2m-1 we have

dYk(Yk-m,Yj)2.

Combining these two inequalities with (4.4), we obtain

dYk(Yj,Yj)dYk(Yk-m,Yk+m)-B0-2
dYk(Yk-m,Yk+m)-C.

This completes the first half of the double induction.

We now know that statements (4.1) and (4.2) hold for all j,j,k with mk-j2m-1 and all j-km. We assume that they hold for mk-jN and j-km for some N2m-1, and prove that they hold for k-j=N+1. The proof of (4.1) is completely analogous to the proof in the first part of the induction, and we omit it. The proof of (4.2) is also similar, but requires one additional step so we give the proof.

We may again write the triangle inequality (4.4). Since m(k-m)-j=N+1-mN, by the inductive hypothesis we have

dYk-m(Yk,Yj)E-CB0,

and so Theorem 2.9 again implies dYk(Yk-m,Yj)B0. If j-k2m-1, then as above dYk(Yk+m,Yj)2. Otherwise, by induction we have

dYk+m(Yk,Yj)E-CB0

and Theorem 2.9 once again implies dYk(Yk+m,Yj)B0. Combining these inequalities with (4.4), we have

dYk(Yj,Yj)dYk(Yk-m,Yk+m)-B0-max{2,B0}
dYk(Yk-m,Yk+m)-C.

This completes the proof of (4.2), and hence the double induction is finished.

Now further assuming (4), we prove (4.3). Note that we must have nm. Without loss of generality we assume that j<k, so that k-j2n2m. For the rest of the proof, for any s,r, sr, we write [s,r]={tstr}.

Suppose that δ is any multi-curve. Let (δ)={s[j,k]i(δ,Ys)0}.

Claim 4.2.

Suppose that s,r[j,k]I(δ). Then |r-s|4n-2.

Observe that by the claim, [j,k](δ) contains fewer than 4n integers.

Proof.

Without loss of generality, we assume that s<r, and suppose for a contradiction that r-s4n-1. By (4), Ys+n,,Ys+3n-1 fills S, and so there exists t with s+nts+3n-1 and t(δ).

Now observe that s+ms+nt and ts+3n-1r-nr-m; by the first part of the theorem we know that

dYt(Ys,Yr)E-C>4.

On the other hand, since i(δ,Ys)=0=i(δ,Yr), and since t(δ) implies πYt(δ), the triangle inequality implies

dYt(Ys,Yr)dYt(Ys,δ)+dYt(δ,Yr)2+2=4,

a contradiction. ∎

Let η be a geodesic in 𝒞(S) connecting Yj to Yk. For any l{j+m,,k-m}, by (4.2) we have that

dYl(Yj,Yk)E-C>G.

Thus Theorem 2.11 guarantees that there is a curve δlη disjoint from Yl. Choose one such δlη for each l[j+m,k-m]. By the previous claim there are at most 4n integers l[j+m,k-m] such that i(δl,Yl)=0, and hence lδl is at most 4n-to-1.

Therefore, η contains at least k-j-2m+14n>k-j4n-m2n curves. It follows that

dS(Yj,Yk)k-j4n-(m2n+1)

proving (4.3). This completes the proof of the theorem. ∎

Theorem 4.3.

Let {Yk}k=0 be an infinite sequence of subsurfaces satisfying conditions (1)(4) in Theorem 4.1. Then there exists a unique νEL(S) so that any accumulation point of {Yk}k=0 in PML(S) is supported on ν.

Proof.

By Theorem 4.1, inequality (4.3), the sequence {Yk}k=0 is (multi-curve) quasi-geodesic in 𝒞(S). Furthermore, 𝒞(S) is δ-hyperbolic. Thus the sequence converges to a point in the Gromov boundary of 𝒞(S). Theorem 2.3 completes the proof. ∎

We complete this section by showing that 𝒫 is sufficient to imply the hypotheses of Theorem 4.1. Given a curve α and an annular subsurface Yβ with core curve β, we note that αYβ if and only if i(α,β)0. Consequently, to remind the reader of the relation to Theorem 4.1, we write αβ to mean i(α,β)0.

Proposition 4.4.

All curves in a sequence {γk}k=0 satisfying P(a) with a>2 and e0E are the core curves of annuli satisfying conditions (1)(4) of Theorem 4.1 with n=m. Consequently, {γk}k=0 is a 1-Lipschitz, (4m,32)-quasi-geodesic in C(S) and there exists νEL(S) so that any accumulation point of {γk}k=0 in PML(S) is supported on ν.

Proof.

Condition (i) of 𝒫 is the same as condition (1) of Theorem 4.1, while (ii) is just condition (4) with n=m. Condition (2) follows from Lemma 3.2. Finally, to see that condition (3) is satisfied, we note that dγk(γk-m,γk+m)ek>akE>2E>E for all km. Furthermore, for k-2m+1jk-m, γjγk by Lemma 3.2, and similarly γjγk, for k+mjk+2m-1. For j and j in these intervals, we obtain i(γj,γk-m)=0 and i(γj,γk+m)=0. Therefore, by the triangle inequality, we have dγk(γj,γj)akE-2>E, as required by (3). ∎

4.1 Subsurface coefficient bounds

We will need estimates on all subsurface coefficients for a sequence satisfying 𝒫. This follows from what we have done so far, together with similar arguments.

Proposition 4.5.

Given a sequence {γk}k=0 satisfying P(a) with a>2 and e0E, then there exists an R>0 with the following properties:

  1. (1)If i,j,k satisfy ji-m and i+mk, then γiγk, γiγj, and
    dγi(γj,γk)+Rei𝑎𝑛𝑑dγi(γj,ν)+Rei.
  2. (2)If WS is a proper subsurface, Wγi for any i, then for any j,k with γjW and γkW,
    dW(γj,γk)<R𝑎𝑛𝑑dW(γj,ν)<R.

Let μ be a marking on S. Then there is a constant R(μ) so that:

  1. For any k sufficiently large and ik-m we have
    dγi(μ,γk)+R(μ)ei𝑎𝑛𝑑dγi(μ,ν)+R(μ)ei.
  2. For any proper subsurface Wγi for any i we have
    dW(μ,γk)<R(μ)𝑎𝑛𝑑dW(μ,ν)<R(μ).

Proof.

We begin with the proofs of (4.5) and (4.6). First note that since any accumulation point of {γk} in 𝒫(S) is supported on ν, any Hausdorff accumulation point of {γk} contains ν. Thus, for any fixed, proper subsurface WS and all sufficiently large k we have πW(ν)πW(γk). Furthermore, since ν is an ending lamination, πW(ν), and hence dW(γk,ν)1, for k sufficiently large. Therefore, for each of (4.5) and (4.6), the statement on the left implies the one on the right after increasing the constant by at most 1. Thus it suffices to prove the two statements on the left.

We begin with (4.5). From the conditions in 𝒫, we have dγi(γi-m,γi+m)=ei. By Theorem 4.1 (which is applicable according to Proposition 4.4), {γl}l=i+mk is a 1-Lipschitz (4m,3/2)-quasi-geodesic such that every curve has nonempty projection to γi. Therefore, by Theorem 2.11 and the triangle inequality we have

|dγi(γi-m,γk)-dγi(γi-m,γi+m)|dγi(γi+m,γk)G.

Note that G depends only on m. Similar reasoning implies

|dγi(γj,γk)-dγi(γi-m,γk)|dγi(γj,γi-m)G.

Combining these, we have

|dγi(γj,γk)-dγi(γi-m,γi+m)|=|dγi(γj,γk)-dγi(γi-m,γk)
+dγi(γi-m,γk)-dγi(γi-m,γi+m)|
2G.

It follows that dγi(γj,γk)+2Gei. For R2G, (4.5) holds.

We now move on to the inequalities in (4.6), and without loss of generality assume that jk. If kj+2m-1, then the conditions in 𝒫 together with Lemma 3.2 imply i(γj,γk)b2, so by Lemma 2.5, dW(γj,γk)2b2+1.

Next, suppose that k=j+2m. Let γk be the element guaranteed by 𝒫, so that

γk=𝒟γk-mek-m(γk).

There are two cases to consider depending on whether γk⋔̸W or γkW. If γk⋔̸W, then since γk=𝒟γk-mek-m(γk)W, we must have γk-mW. Now observe that

jk-m=j+mj+2m-1andk-mkk-m+2m-1.

It follows from the previous paragraph that

dW(γj,γk-m)2b2+1anddW(γk-m,γk)2b2+1,

hence

dW(γj,γk)4b2+2.

Now suppose that γkW. If γk-mW, then just as in the first case we have

dW(γj,γk)4b2+2.

Suppose then that γk-m⋔̸W. If W is not an annulus, then πW(γk)=πW(γk) since 𝒟γk-m is supported outside W. Therefore

dW(γj,γk)=dW(γj,γk)2b2+1

since i(γj,γk)b2. If W is an annulus, because Wγk-m and γk-m⋔̸W, it easily follows that

dW(γj,γk)dW(γj,γk)+dW(γk,γk)(2b2+1)+1

(see e.g. [12]). Therefore, we have shown that if kj+2m, we have

dW(γj,γk)4b2+2.

Now we suppose that k>j+2m. Setting δ=W, as in the proof of Theorem 4.1 we let (δ)={s[j,k]i(δ,γs)0}. Similarly, we let (W)={s[j,k]γsW}, and observe that (δ)(W).

Note that j,k(W), and we let sr be such that [j,s],[r,k](W) are maximal subintervals of (W) containing j and k, respectively (if (W)=[j,k], we can arbitrarily choose js<k and r=s+1 for the argument below). By our choice of r and s, it follows that s+1,r-1(W), and so Claim 4.2 implies r-1-(s+1)4m-2 and hence r-s4m.

Note that since any 2m consecutive curves fill S, either r-s2m, or else there exists s,r(W) such that s<sr<r and r-r,r-s,s-s2m. For example, consider the extremal case that r-s=4m. Then

s=max(W)[s,s+2m]andr=min(W)[s+2m,r]

have the desired properties. Indeed, s-s,r-r are clearly less than 2m. If r-s>2m, then since any 2m consecutive curves fill S, there must be some s<u<r in (W), contradicting the choice of either s or r. The general case is similar.

By the triangle inequality and (4.9) we have

dW(γs,γr)dW(γs,γs)+dW(γs,γr)+dW(γr,γr)12b2+6.

Since {γl}l=js and {γl}l=rk are 1-Lipschitz (4m,3/2)-quasi-geodesics with γlW for all l[j,s][r,k], we can apply Theorem 2.11, and so the triangle inequality and (4.10) give us

dW(γj,γk)dW(γj,γs)+dW(γs,γr)+dW(γr,γk)2G+12b2+6.

So the inequality on the left of (4.6) holds for any R2G+12b2+6. This completes the proof of the first four estimates.

Given a marking μ, note that the intersection number of any curve in μ and any of the curves in the set of filling curves γ0,,γ2m-1 is bounded. Then the estimates in (4.7) follow from the ones in (4.5) and Lemma 2.5 respectively. Similarly the estimates in (4.6) follow from the ones in (4.8). ∎

5 Measures supported on laminations

In this section we begin by proving intersection number estimates for a sequence of curves satisfying 𝒫. Using these estimates, we decompose the sequence into m subsequences and prove that these converge in 𝒫(S). In the next section, we will show that these m limits are precisely the vertices of the simplex of measures on the single topological lamination ν from Proposition 4.4.

5.1 Intersection number estimates

Here we estimate the intersection numbers of curves in the sequence of curves {γk}k=0 satisfying 𝒫. The estimates will be in terms of the constant b and sequence {ek} fixed above. Specifically, given i,k with ki, define

A(i,k):=i+mj<k andjkmodmbej.

When the set of indices of the product is the empty set, we define the product to be 1. It is useful to observe that for ki+2m,

A(i,k)=bek-mA(i,k-m).

It is also useful to arrange the indices as in (3.2) in the following form:

ii+1i+m-1i+mi+m+1i+2m-1i+2mi+2m+1

Then A(i,k) is 1 exactly when k is in the first or second row. If k is below these rows, then the product defining A(i,k) is over all indices j directly above k, up to and including the entry in the second row.

We now state the main estimate on intersection numbers.

Theorem 5.1.

Suppose that {γk}k=0 is a sequence on a surface S satisfying P(a). For a is sufficiently large, there is a constant κ=κ(a)>1, so that for each i,k with ki+m we have

i(γi,γk)*κA(i,k).

Recall that for ik<i+m, i(γi,γk)=0. Combining this with the theorem gives estimates on all intersection numbers i(γi,γk), up to a uniform multiplicative error.

Throughout all that follows, we will assume that a sequence of curves {γk}k=0 satisfies 𝒫=𝒫(a) for a>1.

Outline of the proof

The proof is rather complicated involving multiple induction arguments, so we sketch the approach before diving into the details. The upper bound on i(γi,γk) is proved first, and is valid for any a>1. We start by recursively defining a function K(i,k) for all nonnegative integers ik. By induction, we will prove that

i(γi,γk)K(i,k)A(i,k).

By a second induction, we will bound K(i,k)K1=K1(a), with the bound K1(a) a decreasing function of a. Next, we will recursively define a function K(i,k)=K(i,k,a). By another induction, we prove that

i(γi,γk)K(i,k)A(i,k).

For a sufficiently large, we prove K(i,k,a)K2=K2(a)>0. Setting κ=max{K1,1K2} will prove the theorem.

Upper bound

Recall from 𝒫 (Definition 3.1) that for any k2m, the set of curves {γl}l=k-2mk-1 fill the surface, and the curve γk intersects each of these at most b2 times. Consequently, all complementary components of S(γk-2mγk-1) are either disks or once-punctured disks containing at most 2mb2 pairwise disjoint arcs of γk. In examples we may have many fewer than 2mb2 such arcs, and it is useful to keep track of this constant on its own. Consequently, we set

B2mb2

to be the maximum number of arcs in any complementary component (over all configurations in minimal position).

We are now ready for a recursive definition which will be used in the bounds on intersection numbers (it is useful again to picture the indices as in (5.2)):

K(i,k)={0for ik<i+m,b2for i+mk<i+2m,K(i,k-m)+2Bl=k-2mk-1A(i,l)A(i,k)K(i,l)for i+2mk.

Lemma 5.2.

For all ik, we have i(γi,γk)K(i,k)A(i,k).

The proof takes advantage of the following well-known estimate on the intersection of two curves after applying a power of a Dehn twist on one proved in [13, Exposé 4, Appendix A], see also [19, Section 4, Lemma 4.2].

Proposition 5.3 (Intersection number after Dehn twist).

Let δ, δ, and β be curves in C(S). Then for any integer e

|i(𝒟βeδ,δ)-|e|i(β,δ)i(β,δ)|i(δ,δ).

As above, 𝒟β is a Dehn twist in β. This proposition has the following general application to intersection numbers of curves with the curves in our sequence.

Proposition 5.4.

For any curve δ and any k2m, we have

|i(δ,γk)-bek-mi(δ,γk-m)|2Bl=k-2mk-1i(δ,γl).

Proof.

Since γk=𝒟γk-mek-m(γk), Proposition 5.3 implies

|i(δ,γk)-bek-mi(δ,γk-m)|i(δ,γk).

Assume all curves intersect minimally transversely and that there are no triple points of intersection. From the definition of B, all complementary components of S(γk-2mγk-1) contain at most B pairwise disjoint arcs of γk. Therefore, between any two consecutive intersection points of δ with γk-2mγk-1, there are at most 2B intersections points with γk (any two arcs in a disk component can intersect at most once, and in a once-punctured disk component can intersect in at most two points). Therefore,

i(δ,γk)2Bl=k-2mk-1i(δ,γl).

Combining this with the above inequality proves the proposition. ∎

Proof of Lemma 5.2..

Fix i. The proof is by induction on k. For ik<i+m,

i(γi,γk)=0,K(i,k)=0,A(i,k)=1,

so the lemma follows. Similarly, for i+mk<i+2m, i(γi,γk)b2, K(i,k)=b2, and A(i,k)=1, so again the lemma follows. Now suppose that ki+2m, and assuming that i(γi,γl)K(i,l)A(i,l) for all il<k, we must prove i(γi,γk)K(i,k)A(i,k).

Applying Proposition 5.4 to the case δ=γi, we have

|i(γi,γk)-bek-mi(γi,γk-m)|2Bl=k-2mk-1i(γi,γl).

Therefore, we have

i(γi,γk)bek-mi(γi,γk-m)+2Bl=k-2mk-1i(γi,γl).

Applying the inductive hypothesis and the definitions of A and K to this inequality, we obtain

i(γi,γk)bek-mi(γi,γk-m)+2Bl=k-2mk-1i(γi,γl)
bek-mK(i,k-m)A(i,k-m)+2Bl=k-2mk-1K(i,l)A(i,l)
=A(i,k)K(i,k-m)+A(i,k)2Bl=k-2mk-1A(i,l)A(i,k)K(i,l)
=A(i,k)(K(i,k-m)+2Bl=k-2mk-1A(i,l)A(i,k)K(i,l))
=A(i,k)K(i,k),

as required. ∎

Next we prove that K(i,k) is uniformly bounded, and in particular:

Proposition 5.5.

There exists K1=K1(a)>0 so that for all ik, K(i,k)K1 and in particular, i(γi,γk)K1A(i,k). As a function of a, K1(a) is decreasing.

For the proof of this proposition, we will need the following bound.

Lemma 5.6.

For all il<k, we have

A(i,l)A(i,k)a1-k-im.

Proof.

If k<i+2m, then A(i,l),A(i,k)=1 and a1-k-im1, so the inequality follows.

Now assume that ki+2m. By definition, we have

A(i,l)A(i,k)=i+mj<l andjlmodmbeji+mj<k andjkmodmbej

(where A(i,l) is 1 if l<i+2m). Observe that the denominator has

r=k-(i+m)m=k-im-1>0

terms in the product, indexed by j{k-m,k-2m,,k-rm}, while the numerator has

s=max{0,l-im-1}0

terms, indexed by j{l-m,l-2m,,l-sm} (possibly the empty set). Since l<k, we have sr. Moreover, we have k-pm>l-pm, and thus ek-pm>ael-pm by (3.1), for all p=1,,s. Since (3.1) also implies ej>a for all j1, combining these bounds with the equation above gives

A(i,l)A(i,k)=p=1sel-pmek-pmp=s+1r1ek-pm<p=1sa-1p=s+1ra-1=a-r=a1-k-im,

as required. ∎

As an application, of Lemma 5.6, we prove

Lemma 5.7.

For all ik we have

K(i,k)b2i+mj<k(1+4mBa1-j-i+1m).

As above, the empty product is declared to be 1.

Proof.

The proof is by induction on k. Since K(i,k)b2 for ik<i+2m, the lemma clearly holds for all such k. Now assume that ki+2m, and assume that the lemma holds for all integers less than k and at least i. Let l0 be such that k-2ml0k-1 and

K(i,l0)=max{K(i,l)k-2mlk-1}.

From this, the definition of K(i,k), and from Lemma 5.6 we have

K(i,k)=K(i,k-m)+2Bl=k-2mk-1A(i,l)A(i,k)K(i,l)
K(i,l0)(1+2Bl=k-2mk-1a1-k-im)
=K(i,l0)(1+4mBa1-k-im).

Since l0<k, the proposed bound on K(i,l0) holds by the inductive assumption. Next, observe that the proposed upper bound is an increasing function of k. Indeed, the required bound for K(i,k) is obtained from the one for K(i,k-1) by multiplying by a number greater than or equal to 1. By this monotonicity, the above bound implies

K(i,k)K(i,l0)(1+4mBa1-k-im)
(b2i+mj<k-1(1+4mBa1-j-i+1m))(1+4mBa1-k-im)
=b2i+mj<k(1+4mBa1-j-i+1m).

This completes the proof. ∎

Proof of Proposition 5.5..

The upper bound on K(i,k) in Lemma 5.7 is itself bounded above by the infinite product

K1(a)=b2j=i+m(1+4mBa1-j-i+1m)=b2l=0(1+4mBa-l+1m),

where we have substituted l=j-i-m. We will be done if we prove that this product is convergent, for all a>1, since the product then clearly defines a decreasing function of a.

The infinite product converges if and only if the infinite series obtained by taking logarithms does. Since log(1+x)x, we have

log(b2l=0(1+4mBa-l+1m))=log(b2)+l=0log(1+4mBa-l+1m)
log(b2)+4mBl=0a-l+1m.

The last expression is essentially a geometric series, and hence converges for all a>1, completing the proof. ∎

Lower bound

Let b1 be the constant in 𝒫 (Definition 3.1). We assume a>1 is sufficiently large so that

C=8mBK1j=1a-j<b1

(which is possible since K1=K1(a) is decreasing by Proposition 5.5). For all ki+m, define the function K(i,k) by the following recursive formula for all ki+m:

K(i,k)={Cfor i+mk<i+2m,K(i,k-m)-2Bl=k-2mk-1A(i,l)A(i,k)K(i,l)for i+2mk.

Lemma 5.8.

For all ki+m, we have i(γi,γk)K(i,k)A(i,k).

Proof.

Fix an integer i0. The proof is by induction on k. For the base case, we let i+mk<i+2m. Then A(i,k)=1 and K(i,k)=C<b1, while i(γi,γk)b1, and hence i(γi,γk)K(i,k)A(i,k). We assume therefore that ki+2m and that the lemma is true for all i+ml<k.

Applying Proposition 5.4 to the curve δ=γi, together with Lemma 5.2 and the inductive hypothesis we have

i(γi,γk)ek-mbi(γi,γk-m)-2Bl=k-2mk-1i(γi,γl)
ek-mbK(i,k-m)A(i,k-m)-2Bl=k-2mk-1K(i,l)A(i,l)
=A(i,k)(K(i,k-m)-2Bl=k-2mk-1A(i,l)A(i,k)K(i,l))
=A(i,k)K(i,k),

as required. ∎

Lemma 5.9.

Set K2=C/2>0. Then whenever ki+m, K(i,k)K2.

Proof.

If i+mk<i+2m, then K(i,k)=C>C/2=K2>0. Suppose now that ki+2m, and let k=p+sm, where s and p are positive integers with i+mp<i+2m and pk mod m. Note that

k-im=p+sm-im=s+p-im=s+1.

By Lemma 5.6, it follows that for all l<k, we have A(i,l)A(i,k)a-s. Then from the definition of K and Proposition 5.5 we have

K(i,k)=K(i,k-m)-2Bl=k-2mk-1A(i,l)A(i,k)K(i,l)
K(i,k-m)-2Bl=k-2mk-1a-sK1
K(i,k-m)-2B(2m)a-sK1
=K(i,k-m)-4mBK1a-s.

Iterating this inequality s times implies

K(i,k)K(i,p)-4mBK1q=1sa-q.

Since i+mp<i+2m, K(i,p)=C=8mBK1j=1a-j and hence

K(i,k)4mBK1(2j=1a-j-q=1sa-q)4mBK1j=1a-j=C2=K2.

This completes the proof. ∎

Proof of Theorem 5.1..

For a>1 satisfying (5.4), we have proved that for all ki+m,

K2A(i,k)i(γi,γk)K1A(i,k).

Since K1,K2>0, setting κ=max{K1,1K2} finishes the proof. ∎

Convention

From this point forward, we will assume that 𝒫=𝒫(a) always has a>1 sufficiently large so that (5.4) is satisfied, and consequently the intersection numbers of curves in any sequence {γk}k=0 satisfies (5.3) in Theorem 5.1. For concreteness, we note that from equation (5.4), a16>2 (though in fact, it is much larger).

5.2 Convergence in (S)

Consider again a sequence of curves {γk}k=0 which satisfies the conditions of Theorem 5.1. Let ν(S) be the lamination from Proposition 4.4. In this subsection we will prove this sequence naturally splits into m convergent subsequences in 𝒫(S).

For each h=0,,m-1 and i let

cih=A(0,im+h)=j=1i-1bejm+h,

where A is defined in (5.1).

For each h=0,1,,m-1, define the subsequence γih of the sequence {γk}k=0 by

γih=γim+h.

The main result of this section is the following theorem.

Theorem 5.10.

Suppose that {γk}k=0 satisfies P. Then for each h=0,1,,m-1, there exists a transverse measure ν¯h on ν so that

limiγihcih=ν¯h

in ML(S), where γih and cih are as above.

We will need the following generalization of Theorem 5.1.

Lemma 5.11.

For any curve δ, there exists κ(δ)>0 and N(δ)>0 so that for all kN(δ),

i(δ,γk)*κ(δ)A(0,k).

Remark 5.12.

Note that in Theorem 5.1, we estimate i(γi,γk) with a uniform multiplicative constant κ that works for any two curves γi and γk, but the comparison is with A(i,k) rather than A(0,k). On the other hand, the ratio of A(0,k) and A(i,k) is bounded by a constant depending on i, and not k, so the lemma for δ=γi is an immediate consequence of that theorem.

Proof.

First we note that by Theorem 5.1, we have

i(γi,γk)*κA(i,k).

From the definition of A, and the fact that {ej}j=0 is an increasing sequence, it follows that for each i=0,,2m-1, and all ki, we have the bound

1A(0,k)A(i,k)b2e2me3m.

Setting κ0=κb2e2me3m, for each i=0,,2m-1, we have

i(γi,γk)*κ0A(0,k).

Next, let d=2mκ0. Note that since γ0,,γ2m-1 fills S, the set of measured laminations

Δ={λ¯|j=02m-1i(γj,λ¯)*d1}(S)

is compact. From (5.7) we have

{γkA(0,k)}k=3mΔ.

Let ν(S) be the lamination from Proposition 4.4. Since ν is an ending lamination, the set of measures ν¯Δ supported on ν is a compact subset. By the continuity of the intersection number i, there exists c(δ)>0 so that i(δ,ν¯)*c(δ)1 for all such ν¯.

Let K(δ)(S) be a compact neighborhood which contains the set of measures ν¯ which are supported on ν and are in Δ. By the continuity of the intersection number i again, we can take K(δ) sufficiently small so that there exists κ(δ)>0 such that

i(δ,λ¯)*κ(δ)1for all λ¯K(δ).

Since every accumulation point of {γkA(0,k)}k=3m is a measure ν¯Δ supported on ν, it follows that there exists N(δ) so that

γkA(0,k)K(δ)

for all kN(δ). Consequently, for all kN(δ), we have i(δ,γk)*κ(δ)A(0,k), which completes the proof. ∎

Using the estimates from Lemma 5.11, we prove the next lemma. Theorem 5.10 will then follow easily.

Lemma 5.13.

For any curve δ and any h=0,,m-1, the sequence {i(δ,γihcih)}i=0 converges.

Proof.

By Proposition 5.4 we have

|i(δ,γim+h)-e(i-1)m+hbi(δ,γ(i-1)m+h)|2Bl=(i-2)m+him+h-1i(δ,γl).

Dividing both sides by cih=A(0,im+h)=be(i-1)m+hA(0,(i-1)m+h), and letting κ(δ) be the constant from Lemma 5.11, it follows that for all h=0,,m-1, and i sufficiently large

|i(δ,γim+hcih)-i(δ,γ(i-1)m+hci-1h)|2BA(0,im+h)(l=(i-2)m+him+h-1i(δ,γl))
2BA(0,im+h)(l=(i-2)m+him+h-1κ(δ)A(0,l))
=l=(i-2)m+him+h-12Bκ(δ)A(0,l)A(0,im+h).

Lemma 5.6 implies that the expressions in the final sum admit the following bounds:

A(0,l)A(0,im+h)a1-im+h-0m=a1-i.

Since γih=γim+h, we have

|i(δ,γihcih)-i(δ,γi-1hci-1h)|4mBκ(δ)a1-i.

Consequently, for all i>j sufficiently large, applying this inequality and the triangle inequality we have

|i(δ,γihcih)-i(δ,γjhcjh)|4mBκ(δ)l=j+1ia1-l.

By taking i and j sufficiently large, the (partial) sum of the geometric series on the right can be made arbitrarily small. In particular, {i(δ,γih/cih)} is a Cauchy sequence, hence converges. ∎

Proof of Theorem 5.10..

Fix h{0,,m-1}. Note that since the intersection numbers {i(δ,γih/cih)}i=0 converge for all simple closed curves δ, it follows that {γih/cih}i=0 converges to some lamination ν¯h in the space of measured laminations (S) (since (S) is a closed subset of 𝒞(S)). By Proposition 4.4, ν¯h is supported on ν. ∎

6 Ergodic measures

We continue to assume throughout the rest of this section that {γk}k=0 satisfies 𝒫 and that {γih/cih}i=0 for h=0,,m-1 are the subsequences defined in the previous section limiting to ν¯h supported on ν by Theorem 5.10 for each h=0,,m-1. We say that ν¯h and ν¯h are not absolutely continuous if neither is absolutely continuous with respect to the other one. Note that this is weaker than requiring that the measures be mutually singular.

Recall from the introduction that the space of measures supported on ν is the cone on the simplex of measure Δ(ν). We denote (choices of) the ergodic measures representing the vertices by μ¯0,,μ¯d-1, where 0dξ(S) is the dimension of the space of measure on ν. The ergodic measures are mutually singular since the generic points are disjoint. It follows that if we write ν¯h and ν¯h as nonnegative linear combinations of μ¯0,,μ¯d-1, then ν¯h and ν¯h are not absolutely continuous if and only if there exists μ¯j,μ¯j so that μ¯j has positive coefficient for ν¯h and zero coefficient for ν¯h, while μ¯j has positive coefficient for ν¯h and zero coefficient for ν¯h.

The aim of this section is to show that d=m, and in particular, ν is nonuniquely ergodic. In fact, we will prove that up to scaling and reindexing we have μ¯h=ν¯h.

Using the estimates on the intersection numbers from Theorem 5.1, we first show that the measures ν¯h for h=0,,m-1, are pairwise not absolutely continuous.

Theorem 6.1.

Let h,h{0,,m-1} and hh. Then

limii(γih,ν¯h)i(γih,ν¯h)= and limii(γih,ν¯h)i(γih,ν¯h)=.

In particular, the measures ν¯h and ν¯h are not absolutely continuous with respect to each other.

The last statement is a consequence of the two limits, for if ν¯h and ν¯h were positive linear combinations of the same set of ergodic measures, then these ratios would have to be bounded.

Proof.

For hh, we will calculate that

i(γ0h,γi+1h)i(γih,ν¯h)*1andlimii(γ0h,γi+1h)i(γih,ν¯h)=0.

Dividing the first equation by the second and taking limit (and doing the same with the roles of h and h reversed) gives the desired limiting behavior.

To treat the two estimates in (6.1) simultaneously, we suppose for the time being that h,h{0,,m-1}, but we do not assume hh. From Theorem 5.10 together with (5.5) and (5.6) we have

ν¯h=limkγkhckh=limkγkm+hA(0,km+h).

Combining this with (5.1), (5.6), and the estimate in Theorem 5.1, we see that for any i we may take k sufficiently large so that

i(γ0h,γi+1h)i(γih,ν¯h)*i(γh,γ(i+1)m+h)i(γim+h,γkm+hA(0,km+h))
*A(h,(i+1)m+h)A(im+h,km+h)A(0,km+h).

We will simplify the expression on the right, but the precise formula depends on whether hh or h<h. From the definition (5.1), the right-hand side of (6.2) can be written as

r=1iberm+hr=j0k-1berm+hr=1k-1berm+h=r=1iberm+hr=1j0-1berm+h,

where j0=i+1 if hh and j0=i+2 if h<h. Therefore, from (6.2) we can write

i(γ0h,γi+1h)i(γih,ν¯h)*{r=1ierm+herm+h,hh,1bem+hr=1ierm+he(r+1)m+h,h<h.

Now observe that when h=h, this becomes

i(γ0h,γi+1h)i(γih,ν¯h)*1,

proving the first of the two required equations. So, suppose hh. Then each of the i terms in the product is bounded above by a-1 since the index for the denominator is greater than that of the numerator, and elael-1 for all l1. Thus we have

i(γ0h,γi+1h)i(γih,ν¯h)*a-i,

where when h<h, we have absorbed the constant bem+h into the multiplicative error since m+h<2m. Letting i tend to infinity, we arrive at the second of our required estimates, and have thus completed the proof. ∎

We immediately obtain the following:

Corollary 6.2.

The lamination ν is nonuniquely ergodic.

In fact, Theorem 6.1 implies the main desired result of this section in a special case. To prove this, we first prove a lemma which will be useful in the general case as well.

Lemma 6.3.

If md, then m=d, the measures ν¯0,,ν¯m-1 are distinct and ergodic, and these can be taken as the vertices of Δ(ν).

Proof.

Recall that μ¯0,,μ¯d-1 are ergodic measures spanning the (d-dimensional) space of measures on ν. For each 0h<m, write

ν¯h=j=0d-1cjhμ¯j,

where cjh0 for all j,h. Then for each i, h, and h, we have

i(γih,ν¯h)=j=0d-1cjhi(γih,μ¯j).

Next, fix h and let jh{0,,m-1} be such that cjhh0 and so that there exists a subsequence of γih, so that if 0j<m-1 and cjh0, then

i(γih,μ¯jh)i(γih,μ¯j).

Now suppose that for some hh, cjhh0. On the subsequence of {γih} above where (6.3) holds, Theorem 6.1 implies

=limijcjhi(γih,μ¯j)jcjhi(γih,μ¯j)lim supijcjhi(γih,μ¯j)cjhhi(γih,μ¯jh)
=lim supijcjhcjhhi(γih,μ¯j)i(γih,μ¯jh)jcjhcjhh<.

This contradiction shows that cjhh=0 for all hh. Since cjhh0, it follows that hjh defines an injective function {0,,m-1}{0,,d-1}. Since md, this function is a bijection, m=d, and ν¯h=cjhhμ¯jh. Since μ¯0,,μ¯d-1 are distinct ergodic measures spanning the simplex of measures on ν, the lemma follows. ∎

Corollary 6.4.

If m=ξ(S), then the measures ν¯0,,ν¯m-1 are distinct and ergodic and can be taken as the vertices of Δ(ν).

Proof.

Since the dimension of the space of ergodic measures d is at most ξ(S), it follows that md, and hence Lemma 6.3 implies the result. ∎

6.1 The general case

In [24] Lenzhen and Masur prove that for any nonuniquely ergodic lamination ν the ergodic measures are “reflected” in the geometric limit of a Teichmüller geodesic whose vertical foliation is topologically equivalent to ν. We will use this to prove the following generalization of Corollary 6.4 we need.

Theorem 6.5.

Suppose that {γl}l=0 satisfies P and that {γkh}k=0,h=0,,m-1, is the partition into m subsequences with limkγkh=ν¯h, all supported on ν. Then the measures ν¯0,,ν¯m-1 are distinct and ergodic and can be taken as the vertices of Δ(ν).

Let μ¯0,,μ¯d-1 be the ergodic measures on ν and set

μ¯=j=0d-1μ¯jandγ¯=j=0m-1γj=h=0m-1γ0h.

Here we are viewing the curves in the sum on the right as measured laminations with transverse counting measure on each curve. We choose a normalization for the measures μ¯j so that i(γ¯,μ¯)=1. According to [16], there is a unique complex structure on S from a marked Riemann surface SX and unit area holomorphic quadratic differential q on X with at most simple poles at the punctures, so that the vertical foliation |dx| is μ¯ and the horizontal foliation |dy| is γ¯. Area in the q-metric is computed by integrating dμ¯|dy|. We will also be interested in the measure obtained by integrating dμ¯j|dy| for each j=0,,d-1, which we denote by Areaj. Of course, Area=jAreaj.

Next let g denote the Teichmüller geodesic defined by q. We will write

g(t)=[ft:XX(t)],

where X(t) is the terminal Riemann surface, or

g(t)=[ft:(X,q)(X(t),q(t))],

where q(t) is the terminal quadratic differential. Note that since ν is a nonuniquely ergodic lamination by Masur’s criterion [30] the geodesic g is divergent in the moduli space. The vertical and horizontal measure of a curve γ is denoted vq(t)(γ) and hq(t)(γ), which are precisely the intersection numbers with the horizontal and vertical foliations of q(t), respectively. These are given by

vq(t)(γ)=e-ti(γ,|dy|)=e-ti(γ,γ¯)andhq(t)(γ)=eti(γ,|dx|)=eti(γ,μ¯).

From this it follows that the natural area measure from q(t) is the push forward of the area measure from q. Likewise, this area naturally decomposes as the push forward of the measures Areaj, for j=0,,d-1. Consequently, we will often confuse a subset of X and its image in X(t) and will simply write Area and Areaj in either X or X(t).

Given ϵ>ϵ>0, an (ϵ,ϵ)-thick subsurface of (X(t),q(t)) is a compact surface Y and a continuous map YX(t), injective on the interior of Y with the following properties.

  1. (1)The boundary of Y is sent to a union of q(t)-geodesics, each with extremal length less than ϵ in X(t).
  2. (2)If Y is not an annulus, then every nonperipheral curve in Y has q(t)-length at least ϵ and Y has no peripheral Euclidean cylinders.
  3. (3)If Y is an annulus, then it is a maximal Euclidean cylinder.

Remark 6.6.

We will be interested in the case that ϵϵ. In this case, Y has a large collar neighborhood in Y, which does not contain a Euclidean cylinder (i.e. a large modulus expanding annulus; see [36]). Consequently, Y will have short hyperbolic and extremal length.

As an abuse of notation, we will write YX, although Y is only embedded on its interior. An (ϵ,ϵ)-decomposition of (X(t),q(t)) is a union of (ϵ,ϵ)-thick subsurfaces

Y1(t),,Yr(t)X(t)

with pairwise disjoint interiors. We note that X(t) need not be the union of these subsurfaces. For example, suppose that (X(t),q(t)) is obtained from two flat tori by cutting both open along a very short segment, and gluing them together along the exposed boundary component. If the area of one torus is very close to 1 and the other very close to 0, then an (ϵ,ϵ)-decomposition would consist of the larger slit torus, Y(t), while X(t)-Y(t) would be the (interior of the) smaller slit torus.

The key results from [24] we will need are summarized in the following theorem.

Theorem 6.7 (Lenzhen–Masur).

With the assumptions on the Teichmüller geodesic g above, there exist constants ϵ>0 and B>0 with the following properties. Given any sequence of times tk, there exist a subsequence (still denoted {tk}), a sequence of subsurfaces Y0(tk),,Yd-1(tk) in X(tk), and a sequence ϵk0, so that for all k1:

  1. (1)Y0(tk),,Yd-1(tk) is an (ϵk,ϵ)-thick decomposition,
  2. (2)Areaj(Yj0(tk))>B for all 0jd-1 and for any component Yj0(tk)Yj(tk),
  3. (3)Areaj(Yi(tk))<ϵk for all 0i,jd-1 with ij,
  4. (4)Area(X(tk)-(Y0(tk)Yd-1(tk))<ϵk.

The bulk of this theorem comes from [24, Proposition 1]. More precisely, in [24, proof of Proposition 1], the authors produce a sequence of subsurface {Y(tk)} whose components give an (ϵk,ϵ)-thick decomposition so that each component has area uniformly bounded away from zero, so that the areas of the complements tend to zero. For each ergodic measure μ¯j the authors then find subsurfaces Yi(tk) so that Areaj(Yi(tk))0 as k if ij (see [24, inequality (16)] and its proof). This proves (1), (3), and (4). Since Area=jAreaj, condition (2) follows as well.

To apply this construction, we will need the following lemma. First, for a curve γ and t0, let cylt(γ)X(t) denote the (possibly degenerate) maximal Euclidean cylinder foliated by q(t)-geodesic representatives of γ. We note that cylt(γ)=ft(cyl0(γ)).

Lemma 6.8.

Given any sequence tk, let Y0(tk),,Yd-1(tk)X(tk) denote the (ϵk,ϵ)-thick decomposition from Theorem 6.7 (obtained after passing to a subsequence). Then for all k sufficiently large, each Yj(tk) contains a curve from the sequence {γl} as a nonperipheral curve, or else contains a component which is a cylinder with core curve in the sequence {γl}.

We postpone the proof of this lemma temporarily and use it to easily prove the main result of this section.

Proof of Theorem 6.5..

Let tk be any sequence and let Y0(tk),,Yd-1(tk) be the (ϵk,ϵ)-thick decomposition obtained from Theorem 6.7 after passing to a subsequence. Let k be large enough so that the conclusion of Lemma 6.8 holds. For each j{0,,d-1} let γlj be one of the curves in our sequence so that γlj is either a nonperipheral curve in Yj(tk), or else Yj(tk) contains a cylinder component with core curve γlj. Since Y0(tk),,Yd-1(tk) have disjoint interiors, it follows that γl0,,γld-1 are pairwise disjoint, pairwise nonisotopic curves. By Theorem 5.1, for example, the difference in indices of disjoint curves in our sequence is at most m, and consequently {γl0,,γld-1} consists of at most m curves. That is, md. By Lemma 6.3, d=m, and ν¯0,,ν¯m-1 are ergodic measures spanning the space of all measures on ν, proving the theorem. ∎

6.2 Areas and extremal lengths

The proof of Lemma 6.8 basically follows from the results of [36], together with the estimates on intersection numbers described at the beginning of this section and subsurface coefficient bounds in Section 4.1. Let

g(t)=[ft:(X,q)(X(t),q(t)))]

be the Teichmüller geodesic described above with vertical foliation μ¯=μ¯i, the sum of the ergodic measures on ν, and horizontal foliation |dy|=γ¯.

Suppose that YX(t) is a map of a connected surface into X(t) which is an embedding on the interior, sends the boundary to q(t)-geodesics, and has no peripheral Euclidean cylinders unless Y is itself a Euclidean cylinder (in which case we assume it is maximal). As in the case of thick subsurfaces, we write YX(t), though we are not assuming that Y is thick. Suppose that YX(t) is a subsurface so that the leaves of the vertical and horizontal foliations intersect Y in arcs. This is the case for Y=cylt(γk) for all k sufficiently large, as well as any Y for which ExtX(t)(Y) is small when t is large, and these will be the main cases of interest for us.

As in [36], the surface Y decomposes into a union of horizontal strips

Y=H1(Y)Hr(Y)

and vertical strips

Y=V1(Y)Vr(Y).

Each horizontal strip Hi(Y) is the image of map fiH:[0,1]×[0,1]Y which is injective on the interior, sends [0,1]×{s} to an arc of a horizontal leaf with endpoints on Y. Furthermore, the images of the interiors of f1H,,frH are required to be pairwise disjoint. Let iH=fiH([0,1]×{12}) be a “core arc” of the strip. Vertical strips are defined similarly (and satisfy the analogous properties for the vertical foliation) as are the core arcs 1V,,rV.

Remark 6.9.

This is a slight variation on the strip decompositions in [36].

The width of a horizontal strip Hi(Y), denoted w(Hi(Y)), is the vertical variation of any (or equivalently, every) arc Hi({s}×[0,1]). The width of a vertical strip, w(Vi(Y)), is similarly defined in terms of the horizontal variation. An elementary, but important property of these strips is the following.

Proposition 6.10.

Let YX(t) be as above. If

Y=H1(Y)Hr(Y)=V1(Y)Vr(Y)

is a decomposition into maximal horizontal and vertical strips, then

vq(t)(Y)=2i=1rw(Hi(Y)) and hq(t)(Y)=2i=1rw(Vi(Y)).

The area of Y can be estimated from this by the inequalities

ijw(Hi(Y))w(Vj(Y))(i(iH,jV)-2)
  Area(Y)i,jw(Hi(Y))w(Vj(Y))(i(iH,jV)+2).

To see this, we note that the area of Y is the sum of the areas of the horizontal (or vertical) strips. Every time Vj(Y) crosses Hi(Y), it does so in a rectangle, which contains a unique point of intersection iHjV, except, near the ends of Hi(Y), where we might not see an entire rectangle (and consequently we may or may not see a point of iHjV). We may also have an intersection point in iHjV that does not come in a complete rectangle (but only part of a rectangle). Adding and subtracting 2 to the intersection number accounts for the ends of Hi(Y), and summing gives the bounds.

If Y is nonannular, then note that

i(iH,jV)+2i(πY(γ¯),πY(ν)).

To see this, we note that the horizontal foliation (for example) is γ¯ and πY(γ¯) is basically obtained from the arcs iH by surgering with arcs from the boundary (see also [36, Lemma 3.8]). Combining this inequality with the upper bound in (6.4) and Proposition 6.10, we obtain

Area(Y)hq(t)(Y)vq(t)(Y)i(πY(γ¯),πY(ν)).

Now suppose that Y=cylt(γ) is a maximal Euclidean cylinder with core curve γ. Then there is a decomposition into strips with just one horizontal strip H(Y) and one vertical strip V(Y) and core arcs H and V, respectively. In this case, the intersection number i(H,V) is just dY(γ¯,ν) up to an additive constant (of at most 4 – again, see [36, Lemma 3.8]). Therefore, the bounds in (6.4) together with Proposition 6.10 implies

4Area(cyl0(γ))hq(t)(γ)vq(t)(γ)=4Area(cyl0(γ))i(γ,γ¯)i(γ,μ¯)+dγ(γ¯,ν).

In particular, if dγ(γ¯,ν) is large, then

Area(cyl0(γkh))*hq(t)(γ)vq(t)(γ)dγ(γ¯,ν)=i(γ,γ¯)i(γ,μ¯)γ(γ¯,ν).

The balance time of γ along the Teichmüller geodesic g is the unique t so that

vq(t)(γ)=hq(t)(γ).

Consider Y=cylt(γ)(γ) at the balance time of γ, together with the horizontal and vertical strips H(Y) and V(Y), respectively. In this situation, the rectangles of intersections between H(Y) and V(Y) are actually squares. We can estimate the modulus of Y, which is the ratio of the length to the circumference using these squares. Specifically, we note that the circumference of Y is precisely the length of the diagonal of a square, while the length of Y is approximately half the number of squares, times the length of a diagonal. Since the number of squares is |HV|+dγ(γ¯,ν), we see that the modulus is 2dγ(γ¯,ν), up to a uniform additive error. When dγ(γ¯,ν) is sufficiently large, the reciprocal of this modulus provides an upper bound for the extremal length

Extt(γ)(γ)*1dγ(γ¯,ν).

We note that this estimate was under the assumption that cyl0(γ) was a nondegenerate annulus. In fact, if dγ(γ¯,ν) is sufficiently large (e.g. at least 5), then cyl0(γ) is indeed nondegenerate.

Proof of Lemma 6.8..

Suppose that tk is a sequence of times, Y(tk)X(tk) is a sequence of subsurfaces with q(t)-geodesic boundary, embedded on the interior and having no peripheral Euclidean cylinders, unless Y is itself a Euclidean cylinder in which case we assume it is a maximal Euclidean cylinder. We further assume that ExtX(tk)(Y(tk))0. We pass to a subsequence, also denoted {tk}, and assume that either Y(tk) is nonannular and no nonperipheral curve lies in the sequence {γl}, or that Y(tk) is a cylinder whose core is not a curve from our sequence {γl}. To prove the lemma, it suffices to prove that Area(Y(tk))0, for this implies that such subsurfaces Y(tk) cannot be a component of any Yj(tk) from Theorem 6.7.

Decompose the sequence into an annular subsequence and nonannular subsequence, and we consider each case separately. For the nonannular subsurfaces, we bound the area of Y(tk) using inequality (6.5). Specifically, we note that since no γl is homotopic to a nonperipheral curve in Y(tk), Proposition 4.5 provides a uniform bound for dW(γ¯,ν) for all subsurfaces WY(tk). By Theorem 2.7, follows that i(πY(γ¯),πY(ν)) is uniformly bounded. Since the extremal length of Y(tk) is tending to zero, so is the q(tk)-length, and so also the horizontal and vertical variations:

limkvq(tk)(Y(tk))=0andlimkhq(tk)(Y(tk))=0.

Combining this with (6.5) proves Area(Y(tk))0, as required.

The annular case is similar: Again by Proposition 4.5 since the core curve αk of Y(tk) is not any curve from the sequence {γl}, we have that dαk(γ¯,ν¯) is uniformly bounded, while the horizontal and vertical variations of αk tend to zero (since the extremal length, and hence q(tk)-length, tends to 0). Appealing to (6.6) proves that Area(Y(tk))0 as k in this case, too. ∎

7 Constructions

In this section we provide examples of sequences of curves satisfying 𝒫, and hence to which the results of Sections 36 apply.

7.1 Basic setup

Consider a surface S and m pairwise disjoint, nonisotopic curves γ0,,γm-1. For each k, let Γk=(γ0γm-1)-γk, and let Xk be the component of S cut along Γk containing γk. For each k we assume the following:

  1. (1)Xk contains both γk+1 and γk-1 (with indices taken modulo m),
  2. (2)we have chosen fk:SS a fixed homeomorphism which is the identity on SXk, and pseudo-Anosov on Xk,
  3. (3)the composition of fk and the Dehn twist 𝒟γkr, denoted 𝒟γkrfk, has translation distance at least 16 on the arc and curve graph 𝒜C(Xk) for any r,
  4. (4)there is some b>0 so that i(γk,fk(γk))=b, independent of k.

For 0k,hm-1, let 𝒥(k,h) be the interval from k to h, mod m. This means that if k<h, then 𝒥(k,h)={k,k+1,,h} is the interval in from k to h, while if h<k, then

𝒥(k,h)={k,k+1,,m-1,0,,h}.

If k=h, then 𝒥(k,h)={k}={h}.

For any 0k,hm-1, set

Xk,h=l𝒥(k,h)Xl.

If k=h, note that Xk,h=Xk=Xh. In general, Xk,h is the component of S cut along Γk,h=γh+1γk-1 containing all the curves γk,,γh. That there is such a component follows inductively from the fact that γl±1Xl, with indices taken mod m.

We also define

Fk,h=fkfk+1fh,

where we are composing fl over l𝒥(k,h). Because fl is supported on Xl, it follows that for all 0k,hm-1,

γk,,γh,Fk,h(γh)Xk,h.

In fact, the first and last curves in this sequence fill Xk,h.

Lemma 7.1.

For each 0k,hm-1, {γk,Fk,h(γh)} fills Xk,h. In particular, we have i(γl,Fk,h(γh))0 for all lJ(k,h).

Remark 7.2.

In the case k=h+1 (mod m), we note that Xh+1,h=S and the lemma states that

{γh+1,Fh+1,h(γh)}={γk,fkfk+1fh(γh)}

fills S. We also observe that for all j𝒥(k,h), Xk,jXk,h. It follows that γk,γk+1,,γh and Fk,k(γk),,Fk,h(γh) are contained in Xk,h.

In the following proof, we write πk,h(δ) for the arc-projection to 𝒜C(Xk,h) of a curve δ. This is just the isotopy class of arcs/curves of δ intersected with Xk,h. Likewise, dk,h(δ,δ) is the distance between πh,k(δ) and πh,k(δ) in 𝒜C(Xk,h). We similarly define πk and dk for the case k=h.

Proof.

The last statement follows from the first assertion since, for all l𝒥(k,h), i(γl,γk)=0, and so assuming {γk,Fk,h(γh)} fills, we must have i(γl,Fk,h(γh))0.

The conditions on the curves and homeomorphisms are symmetric under cyclic permutation of the indices, so it suffices to prove the lemma for h=m-1 and 0kh (which is slightly simpler notationally). We write j=h-k and must prove that {γh-j,Fh-j,h(γh)} fills Xh-j,h. We prove this by induction on j.

The base case is j=0, in which case we are reduced to proving that {γh,fh(γh)} fills Xh. This follows from the fact that fh has translation distance at least 16 on 𝒜C(Xh), and hence dh(γh,fh(γh))16.

Suppose that for some 0<jh, {γh+1-j,Fh+1-j,h(γh)} fill Xh+1-j,h, and we must prove that {γh-j,Fh-j,h(γh)} fills Xh-j,h.

Note that since γh-j+1Xh-j, and i(γh-j+1,Fh+1-j,h(γh))0 (because they fill Xh+1-j,h), it follows that Fh+1-j,h(γh) has nontrivial projection to Xh-j. On the other hand, because γh-j is disjoint from Xh+1-j,h (it is in fact a boundary component), it follows that i(γh-j,Fh+1-j,h(γh))=0, hence dh-j(γh-j,Fh+1-j,h)=1. Since fh-j translates by at least 16 on 𝒜C(Xh-j), it follows that

dh-j(Fh-j,h(γh),γh-j)=dh-j(fh-j(Fh+1-j,h(γh)),γh-j)
dh-j(fh-j(Fh+1-j,h(γh)),Fh+1-j,h(γh))
-dh-j(Fh+1-j,h(γh),γh-j)
16-1=15.

Now suppose that {γh-j,Fh-j,h(γh)} does not fill Xh-j,h. Let δ be an essential curve in Xh-j,h which is disjoint from both γh-j and Fh-j,h(γh). Observe that δ cannot intersect the subsurface Xh-j essentially, for otherwise

dh-j(γh-j,Fh-j,h(γh))dh-j(γh-j,δ)+dh-j(δ,Fh-j,h(γh))2

a contradiction.

Therefore, δ is contained in Xh-j,h-Xh-jXh+1-j,h. We first claim that δ must be an essential curve in Xh+1-j,h. If not, then it is contained in the boundary. However, any boundary component of Xh+1-j,h which is essential in Xh-j,h is contained (and essential) in Xh-j. This is a contradiction.

Now since δ is essential in Xh+1-j,h, by the hypothesis of the induction we have

0i(δ,γh+1-j)+i(δ,Fh+1-j,h(γh))=i(δ,γh+1-j)+i(δ,Fh-j,h(γh)).

The last equality follows from the fact that Fh-j,h differs from Fh+1-j,h only in Xh-j, which is disjoint from δ. Finally, we note that γh+1-jXh-j, and hence i(δ,γh+1-j)=0. Consequently,

i(δ,Fh-j,h(γh))0

contradicting our choice of δ. Therefore, {γh-j,Fh-j,h(γh)} fills Xh-j,h. This completes the induction, and hence the proof of the lemma. ∎

Lemma 7.3.

For all 0km-1,

i(γk,Fk,k-1fk(γk))=i(γk,fkfk+1fk-1fk(γk))0.

Proof.

We recall from the previous proof that {γk+1,Fk+1,k(γk)} not only fills S, but satisfies

dk+1(γk+1,Fk+1,k(γk))15.

Since γk+1Xk and γkXk+1 and Xk and Xk+1 overlap, it follows from Theorem 2.9 (see also Remark 2.10) that

dk(γk,Fk+1,k(γk))4.

Since fk translates at least 16 on 𝒜C(Xk), we have

dk(γk,fkFk+1,k(γk))dk(Fk+1,k(γk),fkFk+1,k(γk))-dk(γk,Fk+1,k(γk))
16-412.

Since fkFk+1,k=Fk,k-1fk, the lemma follows. ∎

7.2 General construction

Let {ek}k=0 be a sequence of integers satisfying inequality (3.1) for a>2 sufficiently large as so as to satisfy (5.4) and hence (5.3) in Theorem 5.1 (see the convention at the end of Section 5.1).

For k0, let k¯{0,,m-1} be the residue mod m, and for km define

𝒟k=𝒟γk¯ek-mandϕk=𝒟kfk¯.

The sequence of curves {γk}k=0 is defined as follows:

  1. (1)The first m curves are γ0,,γm-1, as above.
  2. (2)For km, set
    γk=ϕmϕm+1ϕk(γk¯).

Remark 7.4.

We could have avoided having the first m curves as special cases and alternatively defined a sequence {δk}k0 by δk=ϕ0ϕk(γk¯) for all k0. This sequence differs from ours by applying the homeomorphism ϕ0ϕm-1. This is a useful observation when it comes to describing consecutive elements in the sequence, but our choice allows us to keep γ0,,γm-1 as the first m curves.

Proposition 7.5.

With the conditions above, the sequence {γk}k=0 satisfies P for some 0<b1bb2 (where b is the constant assumed from the start).

To simplify the proof, we begin with the following lemma.

Lemma 7.6.

For any 2m consecutive curves γk-m,,γk+m-1, there is a homeomorphism Hk:SS taking these curves to the curves

γk¯,,γk+m-1¯,fk¯(γk¯),,fk¯fk+m-1¯(γk+m-1¯)

(in the same order). Furthermore, the homeomorphism can be chosen to take γk+m to

𝒟fk¯(γk¯)ek(fk¯fk+m-1¯fk¯(γk¯)).

Proof.

We prove the lemma assuming k2m to avoid special cases (the general case can be easily derived from Remark 7.4, for example). We define

Hk=(ϕmϕk-1𝒟k𝒟k+1𝒟k+m-1)-1.

Let h,h{0,,m-1} and note that since i(γh,γh)=0, 𝒟γh(γh)=γh. Furthermore, if hh, from the fact that fh is supported on Xh and γh is disjoint from Xh we easily deduce 𝒟γh and fh commute, and ϕh(γh)=γh.

From these facts we observe that for k-mjk-1, we have

Hk-1(γj)=ϕmϕk-1𝒟k𝒟k+1𝒟k+m-1(γj¯)
=ϕmϕk-1(γj¯)
=ϕmϕj(γj¯)=γj,

while for kjk+m-1, we have

Hk-1(fk¯fj¯(γj¯))=ϕmϕk-1𝒟k𝒟k+m-1fk¯fj¯(γj)
=ϕmϕk-1𝒟kfk¯𝒟jfj¯𝒟j+1𝒟k+m-1(γj¯)
=ϕmϕj𝒟j+1𝒟k+m-1(γj¯)
=ϕmϕj(γj¯)=γj.

This completes the proof of the first statement.

Next, since 𝒟k+m=𝒟γk¯ek, we have

fk¯fk+m-1¯ϕk+m(γk¯)=fk¯fk+m-1¯𝒟k+mfk¯(γk)
=fk¯𝒟k+mfk+1¯fk+m-1¯fk¯(γk¯)
=fk¯𝒟k+mfk¯-1fk¯fk+m-1¯fk¯(γk¯)
=fk¯𝒟γk¯ekfk¯-1fk¯fk+m-1¯fk¯(γk¯)
=𝒟fk¯(γk¯)ekfk¯fk+m-1¯fk¯(γk¯).

Applying Hk-1 to the left-hand side gives γk+m, proving the last statement. ∎

Proof of Proposition 7.5..

Let γk-m,,γk+m-1 be any 2m consecutive curves in our sequence, and let Hk:SS be the homeomorphism from Lemma 7.6 putting these curves into the standard form described by that lemma. Since Hk sends the first m to γk¯,,γk+m-1¯, it follows that these curves are pairwise disjoint. Moreover, the set of all 2m curves fills S by Lemma 7.1 and Remark 7.2 (in fact, the first and last alone fill S). Therefore, the sequence satisfies conditions (i) and (ii) of 𝒫.

To prove that condition (iii) is also satisfied, we need to define γk+m so that

γk+m=𝒟γkek(γk+m),

and verify the intersection conditions. We fix k2m and define

γk+m=ϕmϕm+k-1fk¯(γk¯)

(the case of general km is handled by special cases or by appealing to Remark 7.4). Note that by definition, γk+m=ϕmϕm+k-1ϕm+k(γk¯) and applying Hk to γk and γk+m, Lemma 7.6 gives us

Hk(γk)=fk¯(γk¯)andHk(γk+m)=𝒟fk¯(γk¯)ek(fk¯fk+m-1¯fk¯(γk¯)).

Then, as in the proof of Lemma 7.6 (compare (7.1)), we have

Hk(γk+m)=fk¯fk+m-1¯fk¯(γk¯).

Therefore,

Hk(γk+m)=𝒟Hk(γk)ek(Hk(γk+m))=Hk(𝒟γkek(γk+m)),

so γk+m=𝒟γkek(γk+m).

To prove the intersection number conditions on i(γk+m,γj) from property (iii) of 𝒫, it suffices to prove them for the Hk-images. Thus, for j{k+1,,k+m-1} we note that by Lemma 7.6, Hk(γj)=fk¯fj¯(γj¯), and hence

i(γj,γk+m)=i(fk¯fj¯(γj¯),fk¯fk+m-1¯fk¯(γk¯))
=i(γj¯,fj+1¯fk+m-1¯fk¯(γk¯))
=i(γj¯,γk¯)=0.

The second-to-last equality is obtained by applying (fj+1¯fk+m-1¯fk¯)-1 to both entries, and observing that this fixes γj¯ (cf. the proof of Lemma 7.6).

On the other hand, for j=k, the same basic computation shows

i(γk,γk+m)=i(γk¯,fk¯(γk¯))=b

by assumption (4).

Finally, similar calculations show that for j{k-m,,k-1}, by Lemmas 7.1 and 7.3, we have

i(γj,γk+m)=i(γj¯,fk¯fk+m-1¯fk¯(γk¯))0.

There are only finitely many possible choices of j¯ and k¯, so the values are uniformly bounded between two constants b1<b2. Without loss of generality, we may assume b1bb2. This completes the proof. ∎

While any sequence of curves as above satisfies the conditions in sections in 𝒫 from Definition 3.1, we will need one more condition when analyzing the limits of Teichmüller geodesics. It turns out that any construction as above also satisfies this property. We record this property here for later use.

Lemma 7.7.

Suppose that the sequence {γk}k=0 is constructed as above. If γk,γh are any two curves with mh-k<2m-1, then γk and γh fill a subsurface whose boundary consists entirely of curves in the sequence. Furthermore, for any kjh, γj is either contained in this subsurface, or is disjoint from it. If h-k2m-1, then γk and γh fill S.

Proof.

First assume mh-k2m-1. Applying the homeomorphism Hk:SS from Lemma 7.6, γk and γh are sent to γk¯ and fk¯fh¯(γh¯)=Fk¯,h¯(γh¯), respectively. This fills the surface Xk¯,h¯ which has boundary contained in γ0γm-1. By Lemma 7.1 it follows that Hk-1(Xk¯,h¯) is filled by {γk,γh} and has boundary in Hk(γ0)Hk(γm-1). All the components of this multicurve are in our sequence, as required for the first statement.

For each kjh-m and k+mjh, we hace j¯𝒥(k¯,h¯), and as pointed out in Remark 7.2, γj¯ and Fk¯,j¯(γj¯) are contained in Xk¯,h¯. Consequently, for these values of j, γjHk(Xk¯,h¯). On the other hand, if k<j<h, and j does not fall into one of the above two cases, then h-m+1jk+m-1, which implies 0j-k,h-jm-1 and hence i(γj,γk)=i(γj,γh)=0, and hence γj is disjoint from Hk(Xk¯,h¯). This completes the proof of the second statement.

When h-k=2m-1, we have Xk¯,h¯=S, and hence {γk¯,Fk¯,h¯(γh¯)} fills S. Consequently, {γk,γh} also fills S.

Now we must prove that for h-k2m-1, that γk and γh fill S. The proof is by induction, but we need a little more information in the induction. For simplicity, we assume that km+1 to avoid special cases.

To describe the additional conditions, for k<l, let Φl=ϕmϕl, so that Φk+m-1-1 sends the curves γk,,γh (in order) to the curves

γk¯,,γk+m-1¯,ϕk+m(γk+m¯),,ϕk+mϕh(γh¯).

With this notation, we now wish to prove by double induction (on k and h-k) that for all m+1k<h with h-k2m-1 we have

{γk,γh} fills SanddΦk+m-1(Xk¯)(γk,γh)12.

The base case is h-k=2m-1 and any km+1. We have already pointed out that {γk,γh} fills S. We note that applying Φk+m-1 takes γk+1,,γh to

γk+1¯,,γk+m¯,ϕk+m+1(γk+m+1¯),,ϕk+mϕk+2m-1(γk+2m-1¯).

For the first and last curves {γk+1¯,ϕk+m+1ϕk+2m-1(γk+2m-1¯)} we see that these fill

Xk+1¯,k+2m-1¯=Xk+1¯,k-1¯

which has γk¯ as a boundary component. Since πk¯(ϕk+m+1ϕk+2m-1(γk+2m-1¯)) is disjoint from γk¯, it follows that applying ϕk+m to this last curve ϕk+m+1ϕk+2m-1(γk+2m-1¯) we have

dk¯(γk¯,ϕk+mϕk+m+1ϕk+2m-1(γk+2m-1¯))14>12.

But notice that Φk+m-1-1(γk+2m-1)=ϕk+mϕk+2m-1(γk+2m-1¯) while on the other hand Φk+m-1-1(γk)=γk¯, hence

dΦk+m-1(Xk¯)(γk,γk+2m-1)12,

as required for the base case.

For the induction step, the proof is quite similar. We assume that the statement holds for all km+1 and all 2m-1h-kN, and prove it for h-k=N+1. Since h-(k+1)=N and k+1m+2m+1, by the inductive assumption it follows that {γk+1,γh} fills S and that

dΦk+m(Xk+1¯)(γk+1,γh)12.

Therefore, applying Φk+m-1, we have

dk+1¯(γk+1¯,ϕk+m+1ϕh(γh¯))12.

The homeomorphism Φk+m-1 sends γk,,γh to the sequence

ϕk+m-1(γk¯),γk+1¯,,γk+m¯,ϕk+m+1(γk+1¯),,ϕk+m+1ϕh(γh¯).

Since γk¯Xk+1¯ and γk+1¯Xk¯, Theorem 2.9 (see also Remark 2.10) ensures that we have

dk¯(γk¯,ϕk+m+1ϕh(γh¯))4.

Applying ϕk+m (which translates by at least 16 on 𝒞(Xk¯)) to the second curve, we get

dk¯(γk¯,ϕk+mϕk+m+1ϕh(γh¯)12.

In particular, we have

dΦk+m-1(Xk¯)(γk,γh)12.

This proves part of the requirement on γk,γh.

We must also show that {γk,γh} fills the surface S. We will show that the Φk+m-1-image {γk¯,ϕk+mϕh(γh¯)} fills S, which will suffice. To see this, take any essential curve δ and suppose it is disjoint from both γk¯ and ϕk+mϕh(γh¯). Then note that δ must have empty projection to Xk¯, for otherwise the triangle inequality implies that the distance from πk¯(γk¯) to πk¯(ϕk+mϕh(γh¯)) is at most 4, a contradiction to the fact that

dk¯(γk¯,ϕk+m+1ϕh(γh¯))=dΦk+m-1(Xk¯)(γk,γh)12.

Since {γk+1¯,ϕk+m+1ϕh(γh¯)} fills S, δ must intersect one of these curves. However, γk+1¯ is contained in the boundary of Xk¯, and hence δ is disjoint from this. Consequently, δ must intersect ϕk+m+1ϕh(γh¯). Since ϕk+m is supported on Xk¯ which is disjoint from δ, we have

0i(δ,ϕk+m+1ϕh(γh¯))=i(ϕk+m-1(δ),ϕk+m+1ϕh(γh¯))
=i(δ,ϕk+mϕh(γh¯)).

This contradicts our initial assumption on δ, hence no such δ exists and {γh¯,ϕk+mϕh(γh¯)} fills S as required. This completes the proof. ∎

7.3 Specific examples

Here we provide two specific families of examples of the general construction, but it is quite flexible and easy to build many more examples. We need to describe γ0,,γm-1, together with the rest of the data from the beginning of Section 7.1. For this, we will first ensure that all of our subsurfaces Xk have the property that γk±1Xk (indices mod m). This is the first of the four conditions required. For the other three conditions, it will be enough to choose the sequence so that for any 0k,hm-1, there is a homeomorphism of pairs (Xk,γk)(Xh,γh). For then, we can choose f0:SS any homeomorphism which is the identity on SXk, pseudo-Anosov on 𝒜C(Xk) with translation distance at least 15, and then use the homeomorphisms (X0,γ0)(Xk,γk) to conjugate f0 to homeomorphisms fk:SS.

7.3.1 Maximal-dimensional simplices

For the first family of examples, we can choose a pants decomposition on Sg,0 a closed genus g3 surface as shown in Figure 2. Each Xk is homeomorphic to a 4-holed sphere, and γkXk is an essential curve. Any two (Xk,γk) and (Xh,γh) are clearly homeomorphic pairs. In this case m=3g-3, and the limiting lamination ν from Proposition 4.4 defines a simplex of measures with maximal possible dimension in 𝒫(S) by Theorem 6.5. One can also construct examples in genus 2 by taking γ0,γ1,γ2 to be a pants decomposition of non-separating curves.

Figure 2
Figure 2

The pairwise disjoint curves γ0,,γm-1 for the first family of examples in the case of genus 5 (and hence m=12).

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0024

7.3.2 Non-maximal examples

For our second family, we choose m=g-1, and take a sequence γ0,,γm-1 as shown in Figure 3. Here each Xk is homeomorphic to a surface of genus 2 with two boundary components and γk is a curve that cuts Xk into two genus 1 surfaces with two boundary components.

Figure 3
Figure 3

The pairwise disjoint curves γ0,,γm-1 for the second family in the case of genus 5 (and hence m=4).

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0024

8 Teichmüller geodesics and active intervals

In [36, 37, 38] the fourth author has developed techniques to control the length-functions and twist parameters along Teichmüller geodesics in terms of subsurface coefficients. In [22] this control was used to study the limit sets of Teichmüller geodesics in the Thurston compactification of Teichmüller space. Here we also appeal to this control. Most of the estimates in this section are similar to the ones in [22, Section 6].

For the remainder of this section and the next we assume that {γk}k=0 is a sequence of curves satisfying the condition 𝒫 from Definition 3.1 with a>1 large enough to satisfy (5.4) and consequently so that (5.3) in Theorem 5.1 holds, and the sequence of powers {ek}k=0 satisfy the growth condition (3.1) for this a. For h=0,,m-1, let γih=γim+h, as usual.

Let ν be the nonuniquely ergodic lamination determined by the sequence (see Theorem 4.3 and Corollary 6.2). Furthermore let ν¯h, for h=0,,m-1, be the ergodic measures from Theorems 5.10 and 6.5, so that γihν¯h in 𝒫(S), for each h. Let

ν¯=h=0m-1xhν¯h,

for any xh>0 for each h=0,,m-1.

Let XTeich(S) and μ be a short marking at X. By [18], there is a unique Teichmüller geodesic ray starting at X with vertical foliation ν¯, and we let η¯ be the horizontal foliation (with support η). Denote the Teichmüller geodesic ray by r:[0,)Teich(S). For a t, we sometimes denote r(t)=Xt and denote the quadratic differential at Xt by qt. We write vt(α),ht(α),t(α) for the qt-vertical variation, qt-horizontal variation, and qt-length of α, respectively. In particular,

vt(α)=exp(-t)i(α,η¯),ht(α)=exp(t)i(α,ν¯),t(α)*vt(α)+ht(α).

We write Hypt(α)=HypXt(α), the Xt-hyperbolic length of α and wt(α)=wXt(α) for the Xt-width, and recall from (2.1) that

wt(α)+2log(1Hypt(α)).

We also recall that ϵ0>0 is the Margulis constant, and that any two hyperbolic geodesics of length at most ϵ0 must be embedded and disjoint.

For any curve α let cylt(α) be the maximal flat cylinder foliated by all geodesic representatives of α in the qt metric, as in Section 6.1, and let mod(cylt(α)) denote its modulus. Fix M>0 sufficiently large so that for any curve α with mod(cylt(α))M, for some t, then Hypt(α)ϵ0. For any k, let Jγk, also denoted Jk, be the active interval of γk

Jk={t[0,)mod(cylt(γk))M}.

Write Jk=[a¯k,a¯k] and denote the midpoint of Jk by ak (the balance time of γk along the geodesic, i.e. the unique t when vt(γk)=ht(γk)). For each h{0,,m-1} and i0, we also write Jim+h=Jih, aih=aim+h, a¯ih=a¯im+h, and a¯ih=a¯im+h, to denote the data associated to γih=γim+h.

Proposition 8.1 (Active intervals of curves in the sequence).

With the assumptions and notation as above, we have the following:

  1. (i)For k sufficiently large, Jk. Moreover, JkJl= whenever i(γk,γl)0.
  2. (ii)For 0f<k sufficiently large with k-fm, Jf occurs before Jk. Consequently, some tail of each subsequence {Jih}i=0 appears in order.
  3. (iii)For k sufficiently large and a multiplicative constant depending only on ν and X,
    Hypak(γk)*1dγk(μ,ν)*1ek.
  4. (iv)For an additive constant depending only on ν, X, and M, we have
    |Jk|+logdγk(μ,ν)+log(ek).

The following will be convenient for the proof of Proposition 8.1.

Lemma 8.2.

With notation and assumptions above, there exists k00 sufficiently large so that if YS is a subsurface such that for some kk0, dS(γk,Y)2, then

dY(μ,ν)+G+1dY(η,ν),

where G is the constant from Theorem 2.11 (for a geodesic).

Proof.

Let g be a geodesic in 𝒞(S) from (any curve in) μ limiting to η if η is an ending lamination, or from μ to any curve α with i(α,η¯)=0 otherwise. Since η and ν fill S, and γkν𝒞(S), the distance from γk to g tends to infinity with k. For Y and γk as in the statement of the lemma, dS(Y,γk)2, and hence for k sufficiently large, Y has distance at least 4 from g. Consequently, Y intersects every curve on g, and Theorem 2.11 guarantees that diamY(g)G. Thus for all βg, dY(β,μ)G. Since g limits to η (or one of it is curves is disjoint from η), it follows that dY(η,μ)G+1, and so the lemma follows from the triangle inequality in 𝒞(Y). ∎

Proof of Proposition 8.1..

From [36], if dγk(η,ν) is sufficiently large, then at the balance time ak, cylak(γk) has modulus at least M. For all k sufficiently large, (4.7) and Lemma 8.2 imply

dγk(η,ν)+dγk(μ,ν)+ek.

By construction, ek as k, and hence Jk for all sufficiently large k. Furthermore, for all tJk, we have Hypt(γk)ϵ0. Since two curves with length bounded by ϵ0 are disjoint, part (i) follows.

By (4.5) in Proposition 4.5 we have for all 0f<k<l with l-k,k-fm that

dγk(γf,γl)+ek.

Let N0 be such that for all kN, ek>B0, where B0 is the constant from Proposition 2.9. Thus for all N<f<k<l with l-k,k-fm we have

dγf(γk,γl)B0.

Since γkν𝒞(S), the triangle inequality in 𝒞(γk) implies that

dγf(γk,ν)+0

for all Nf<k with k-fm. Let N0N be sufficiently large so that if fN0, then dγf(η,ν)+ef. Thus, for k-fm, fN0, at the balance time t=af of γf, the qt-geodesic representative of γk is more vertical than horizontal, and hence af<ak. By part (i), the intervals Jf and Jk are disjoint, so part (ii) holds. (See also the discussion in [36, Proposition 5.6].)

For part (iv), observe that by [36], the modulus of cylt(γk) satisfies

mod(cylt(γk))*dγk(η,ν)cosh2(t-ak).

For k is sufficiently large, Lemma 8.2 implies dγk(η,ν)+dγk(μ,ν)+ek. At the endpoint a¯k of Jk, mod(cyla¯k(γk))=M. Since |Jk|=2(a¯k-ak), we have

M*ekcosh2(12|Jk|).

Taking logarithms we obtain log(ek)-|Jk|+log(M), proving part (iv).

We proceed to the proof of part (iii). Following Rafi in [36, Section 6], we introduce the following constants associated to a curve α𝒞(S) and an essential subsurface YS with αY (when Y is an annulus, recall that αY means that α is the core curve of Y).

  1. If Y is a nonannular subsurface, an arc β in Y is a common K-quasi-parallel of πY(η) and πY(ν) for α and Y if β transversely intersects α and
    max{i(β,πY(η)),i(β,πY(ν))}K.
    Here πY(η) denotes the arc-and-curve projection of η: the union of arcs and curves obtained by intersecting η with Y (likewise for ν). Define K(Y)=logK, where K is the smallest number so that η and ν have a common K-quasi-parallel.
  2. If Y is an annular subsurface, let K(Y)=dY(η,ν).

Now define Kα to be the largest K(Y) where αY. Then [36, Theorem 6.1] implies that

Hypa(α)*1Kα,

where a is the balance time of α along the geodesic ray r.

In what follows we show that for all sufficiently large k, Kγk is approximately equal to ek. Since we will be interested in subsurfaces Y with γkY (or subsurfaces of those, ZY), we can apply to Lemma 8.2 deducing that

dY(η,ν)+dY(μ,ν).

We will assume that k is sufficiently large for this to hold, and will use this without further mention.

First suppose Y is the annulus with core curve γk, and observe that by Proposition 4.5 and Lemma 8.2,

dY(η,ν)+dY(μ,ν)+ek,

thus K(Y)+ek. So we consider the case that Y is a nonannular subsurface with γkY, and prove that for sufficiently large k, K(Y)ek.

If Y contains no curves γk from the sequence as essential curves, then for every subsurface ZY, by Proposition 4.5 and Lemma 8.2 we have

dZ(η,ν)+dZ(μ,ν)+0.

Then choosing the threshold A in Theorem 2.7 larger than the upper bound on these projections, and applying the theorem to πY(η),πY(ν), we see that

i(πY(η),πY(ν))+0.

In this case we have K(Y)+0, and so K(Y)ek for all sufficiently large k.

Next we suppose that there are curves from our sequence contained in Y. Let

{γl}l{γf}f=0,

where is an ordered subset of which is the set of curves from our sequence which are contained in Y. From (4.1) in Theorem 4.1 we see that {k-m+1,,k+m-1} since any other curve in the sequence intersects γk. We proceed to find an upper bound for the factor K(Y). For this purpose let βπY(γk+m) be any component arc of the projection. Then from Theorem 2.7 and Lemma 8.2 we have

i(β,πYν)WY,nonannular{dW(γk+m,ν)}A+WY,annularlog{dW(γk+m,ν)}A

and

i(β,πYη)WY,nonannular{dW(γk+m,η)}A+WY,annularlog{dW(γk+m,η)}A
WY,nonannular{dW(γk+m,μ)}A+WY,annularlog{dW(γk+m,μ)}A.

Choose the threshold constant A from Theorem 2.7 larger than the constant R(μ) from Proposition 4.5. Appealing to that proposition and the fact that any l is less than k+m, the first of these equations implies that i(β,πYν)0. For the second set of equations, note that any l with γlγk+m has lk. Therefore, by Theorem 2.7 and the fact that {ef} is increasing, we have

i(β,πYμ)llog{dγl(γk+m,μ)}A
l=k-m+1klog(dγl(γk+m,μ))
l=k-m+1klog(el)mlog(ek)ek.

Therefore, β is a K-quasi-parallel with Kek. Consequently,

K(Y)log(K)log(ek)ek.

This completes the proof of part (iii), and hence the proposition. ∎

Next we list some estimates for the locations of the intervals Jih[0,), and provide more information on the relative positions of the intervals.

Let h{0,,m-1}. From part (i) and (iv) of Proposition 8.1, together with the definitions, we have that for i sufficiently large

a¯ih+aih-logeih2,
a¯ih+aih+logeih2.

Together with these estimates, the next lemma tells us the location of the active intervals, up to an additive error.

Lemma 8.3.

For any h={0,,m-1} and i sufficiently large,

aih+j=0i-1logbejh+logeih2-logxh2.

The additive error depends on X, γ0h, and ν.

Proof.

The proof of this lemma is similar to that of [22, Lemma 6.3], so we just sketch the proof. Choose i sufficiently large so that Jih and aih>0, and so that we may estimate i(γih,μ) using Lemma 5.11 (since μ is a finite set of curves). Then appealing to the fact that X is a fixed surface and μ a short marking, we have

v0(γih)*l0(γih)*Hyp0(γih)*i(γih,μ)*A(0,h+im)=j=0i-1bejh.

Since vt(γih)ht(γih) is constant in t, and vaih(γih)=haih(γih), we have, for i sufficiently large,

vaih2(γih)=vaih(γih)haih(γih)
=v0(γih)h0(γih)
*i(γih,μ)i(γih,ν¯)
*i(γih,μ)(d=0m-1xdi(γih,ν¯d)).

Since μ is a fixed set of curves and γ0h a fixed curve, i(γ0h,γih)*i(μ,γih) for all i sufficiently large. Thus from (6.1), for hd, d{0,,m-1}, we have

i(γih,ν¯h)*1i(γi+1h,μ)andi(γih,ν¯d)i(γi+1h,μ)0.

The above estimates and Lemma 5.11 imply that for i sufficiently large,

vaih2(γih)*xhi(γih,μ)i(γi+1h,μ)*xhbeih.

Combining this with (8.5), we have

exp(aih)=v0(γih)exp(-aih)v0(γih)=v0(γih)vaih(γih)*j=0i-1bejhxhbeih.

Solving for aih and taking logarithms (discarding a constant logb) proves (8.4), completing the proof. ∎

Lemma 8.4.

For any k sufficiently large, we have a¯k+a¯k+m, with additive error depending on X, M, γ0h, and ν.

Proof.

Let k=im+h, where h{0,,m-1}. From (8.2), (8.3) and (8.4) we calculate

a¯k+m-a¯k=a¯i+1h-a¯ih
+j=0ilogbejh+logei+1h2-logxh2-logei+1h2
-(j=0i-1logbejh+logeih2-logxh2+logeih2)
=logbeih-logeih=logb.

Therefore a¯k+a¯k+m since logb is a constant. ∎

Let k,l and 0<l-km. Suppose that khmodm and ldmodm, where h,d{0,,m-1}. Then for the pair (k,l) one of the following two hold:

h<d and there exists an i, so that k=mi+h and l=mi+d,
h>d and there exists an i, so that k=mi+h and l=m(i+1)+d.

Notation 8.5.

Let {xi}i=0 and {yi}i=0 be sequences of real numbers. We write xiyi if xi<yi for all i sufficiently large and yi-xi as i.

Lemma 8.6.

For k,lN sufficiently large, where 0l-k<m, the following holds:

a¯k-m<a¯la¯k.

Proof.

The proof is similar to the proof of [22, Lemma 7.3]. For the first inequality, note that l-(k-m)m. By Proposition 8.1 (i)–(ii), Jk-m occurs before Jl, and so we have a¯k-m<a¯l.

We show that a¯la¯k. If l=k, then since |Jk| as k, we have a¯ka¯k. Now assume that k<l and let khmodm and ldmodm with h,d{0,,m-1}. First, suppose that (8.6) holds so h<d. Using (3.1), (8.2), (8.3) and (8.4), and the fact that ekak-fef for k>f, we have

a¯k-a¯l=a¯ih-a¯id
+j=0i-1logbejh+logeih-12logxh-j=0i-1logbejd+12logxd
=j=0i-1logejhejd+logeih+12logxdxh
=j=1ilogejhej-1d+loge0h+12logxdxh
j=1i(m+h-d)loga+12logxdxh
=i(m+h-d)loga+12logxdxh.

Now since m+h-d>0, the last term goes to as i.

Next suppose that (8.7) holds so h>d. Then we similarly have

a¯k-a¯l=a¯ih-a¯i+1d
+j=1i-1logbejh+logeih-j=1ilogbejd+12logxdxh
=j=1ilogejhejd+12logxdxh-logb
=j=1ilogejhejd+12logxdxh-logb
=i(h-d)loga+12logxdxh-logb.

Now since h-d>0, the last term goes to as i. ∎

To obtain a greater control over the arrangement of intervals Jk along the Teichmüller geodesic ray (see Lemma 8.8 below) we consider the following growth conditions, in addition to (3.1):

ek+1(j=0kej)2.

Such sequences exist simply by setting e0a and defining ek recursively, ensuring at every step that (8.8) is satisfied.

Condition (8.8) has the following consequence.

Lemma 8.7.

Suppose that a sequence {ek}k satisfies (3.1) and (8.8).

  1. (i)If (8.6) holds, then
    (eid)12eihj=0i-1ejdejh.
  2. (ii)If (8.7) holds, then
    (ei+1d)12j=0iejdejh.

Proof.

Let kdmodm and lhmodm, where d,h{0,,m-1}. First suppose that (8.6) holds so h<d. Since {ek} is increasing (more than) exponentially fast, we have

j=0i-1ejdejh.

Moreover, by (8.8) we have

(eid)12eih,

that is,

(eid)12eih1.

Thus (i) follows.

Now suppose that (8.7) holds so h>d. Then

(ei+1d)12j=0m(i+1)+d-1ejj=0iejh,

where the second inequality holds because m(i+1)+d>mi+h. Therefore, condition (ii) easily follows in this case as well. ∎

Lemma 8.8.

Suppose that the growth condition (8.8) holds. Then for k,lN sufficiently large with 0<l-k<m we have

a¯kal.

Proof.

Let fhmodm and ldmodm, where h,d{0,,m-1}.

First suppose that (8.6) holds so h<d. Then from (8.3) and (8.4) we calculate

al-a¯k=aid-a¯ih
+j=0i-1logbejd+logeid2-logxd2-(j=0i-1logbejh+logeih-logxh2)
=log((eid)12eihj=0i-1ejdejh)+12logxhxd,

where the sequence tends to infinity as i by Lemma 8.7.

Now suppose that (8.7) holds so h>d. Then we have

al-a¯k=ai+1d-a¯ih
+j=0ilogbeid+logei+1d2-logxd2-(j=0i-1logbejh+logeih-logxh2)
=log((ei+1d)12j=0iejdejh)+logb+12logxhxd,

where again the convergence to infinity as i is by Lemma 8.7. ∎

The following conveniently summarizes the relative positions of intervals for large indices. See Figure 4.

Lemma 8.9.

For k<l sufficiently large and l<k+m, we have

a¯ka¯laka¯k<a¯k+mala¯l<a¯l+mak+m.

Furthermore,

a¯k+a¯k+m.

Proof.

This is immediate from Lemmas 8.4, 8.6 and 8.8. ∎

Figure 4
Figure 4

Relative positions of active intervals, k<l<k+m.

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0024

9 Limit sets of Teichmüller geodesics

In this section, we continue with the assumptions from the previous section on the sequences {γk}k=0 and {ek}k=0 (including both condition (3.1) and condition (8.8)), limiting lamination νC(S) of {γk}k=0, Teichmüller geodesic ray r(t)=Xt with quadratic differential qt at time t[0,), vertical foliations ν¯=h=0m-1xiν¯h and horizontal foliation η¯ for (X,q)=(X0,q0), short marking μ for X, and active intervals Jk=[a¯k,a¯k] with midpoint ak. We will also be appealing to all the estimates from the previous sections regarding this data.

In addition, we will need one more condition on {γk}k=0, which we add to the properties 𝒫 assumed already: For any k0, let

σk=γkγk+1γk+m-1.

The additional condition is

  1. $\mathcal{P}$ (iv)Let α be any essential curve in S\σk. Then there is no subsurface YS with αY which is filled by a collection of the curves in the sequence {γk}k=0.

Recall that when Y is an annular subsurface by αY, we mean that α is the core curve of Y.

Remark 9.1.

Note that when σk is a pants decomposition of S, condition $\mathcal{P}$ (iv) holds vacuously because there are no essential curves in S\σk. Together with the other conditions in 𝒫, the new condition $\mathcal{P}$ (iv) is equivalent to requiring that any subsurface filled by a subset of {γk}k=0 has as boundary a union of curves in {γk}k=0. According to Lemma 7.7 condition $\mathcal{P}$ (iv) holds for the sequences constructed in Section 7.

Under these assumptions, Theorem 1.4 from the introduction, which describes the limit set of r(t) in the Thurston compactification Teich¯(S)=Teich(S)𝒫(S), can be restated as follows. Recall that the set of projective classes of measures on ν is a simplex Δ(ν) spanned by the projective classes of the ergodic measures [ν¯0],,[ν¯m-1].

Theorem 9.2.

The accumulation set of r(t) in PML(S) is the simple closed curve in the simplex Δ(ν) that is the concatenation of edges

[[ν¯0],[ν¯1]][[ν¯1,ν¯2]][[ν¯m-1],[ν¯0]].

We begin by reducing this theorem to a more manageable statement (Theorem 9.3), which also provides more information about how the sequence limits to the simple closed curve. We then briefly sketch the idea of the proof, and describe some of the necessary estimates. After that we reduce the theorem further to a technical version (Theorem 9.17), providing even more detailed information about what the limit looks like, and which allows for a more concise proof. After supplying the final estimates necessary, we carry out the proof.

9.1 First reduction and sketch of proof

By Proposition 8.1, the intervals Jk are nonempty for all k sufficiently large. Combining this with Lemma 8.8, it follows that for all k<l sufficiently large, a¯k<a¯l, and that a¯l with l. Therefore, the set of intervals [a¯k,a¯k+1] for all sufficiently large k, cover all but a compact subset of [0,), and consecutive segments intersect only in their endpoints. Theorem 9.2 easily follows from

Theorem 9.3.

Fix h,h{0,,m-1} with hh+1modm and suppose that {ti} is a sequence with ti[a¯im+h,a¯im+h+1] for all sufficiently large i. Then r(ti)=Xti accumulates on the edge [[ν¯h],[ν¯h]]Δ(ν).

Furthermore, if {ti-a¯im+h} is bounded independent of i, then

limiXti=[ν¯h].

Proof of Theorem 9.2 assuming Theorem 9.3..

From the second part of Theorem 9.3 applied to ti=a¯im+h, it follows that for all h{0,,m-1<