## 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 *X* in Teichmüller space, the Teichmüller ray based at *X* and defined by the quadratic differential with vertical foliation

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 *m**-**local* for some
*m* as the *subsequence counter*. Most of the conditions in

*For appropriate choices of parameters in *

*If *

*On the other hand, there is a constant *

See Propositions 4.4 and 4.5 for precise statements. Here *W* and

Although the conditions in *m*. There are *m* subsequences

*For appropriate choices of parameters in *

*m*is the subsequence counter, then the dimension of the space of measures on ν is precisely

*m*, and the

*m*subsequences

*m*ergodic 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 *ergodic measures*, and the dimension of the space of measures is at most *m* can also be at most

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:

*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

*Suppose that ν is the limiting lamination of a sequence *

*where *

*r*in the Thurston boundary is the simple closed curve in the simplex

When

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

For

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 *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.

Let

- •
if$f{\stackrel{+}{\asymp}}_{C}g$ for all$f\left(x\right)-C\le g\left(x\right)\le f\left(x\right)+C$ ,$x\in X$ - •
if$f{\stackrel{*}{\asymp}}_{K}g$ for all$\frac{1}{K}f\left(x\right)\le g\left(x\right)\le Kf\left(x\right)$ ,$x\in X$ - •
if$f{\asymp}_{K,C}g$ for all$\frac{1}{K}\left(f\left(x\right)-C\right)\le g\left(x\right)\le Kf\left(x\right)+C$ ,$x\in X$ - •
if$f{\stackrel{*}{\prec}}_{K}g$ for all$f\left(x\right)\le Kg\left(x\right)$ ,$x\in X$ - •
if$f{\stackrel{+}{\prec}}_{C}g$ for all$f\left(x\right)\le g\left(x\right)+C$ ,$x\in X$ - •
if$f{\prec}_{K,C}g$ for all$f\left(x\right)\le Kg\left(x\right)+C$ .$x\in X$

When the constants are known from the text we drop them from the notations. Finally, we also write

Let *S* by
*S* with

### 2.1 Curve complexes

For any surface *Y*, *Y*, denoted by *Y* that are essential, meaning non-null homotopic and nonperipheral. For *k*-simplex if any pair can be represented by disjoint curves. For *Y* is *k*-simplex if the curves can be represented intersecting twice (for

The only surface *Y* with *S*, let *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

Distances between vertices in *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

For surfaces *Y* with

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

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

When the number *S* are punctures, *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

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

### 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

*There is a homeomorphism Φ from the Gromov boundary of
*

*Let *

*x*in the Gromov boundary of

We will use this theorem throughout to identify points in

### 2.4 Subsurface coefficients

An *essential subsurface**Y* of a surface *Z* with *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 *Y* is the compactified annular covers *Z* associated to a simple closed curve α in *Z*. We sometimes confuse an annular neighborhood of α with the cover *Z*.

Let *Y*. Represent *Y* as a component of the complement of a very small neighborhood of geodesic multicurve. If

For a marking μ (or a partial marking), if *Y*,

Let *Z* and *Y**-**subsurface coefficient* of μ and

The subsurface coefficient is sometimes alternatively defined as the (minimal) distance between

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

*Given curves *

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

The bound in the above lemma can be improved to

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

*Given *

*where W is so that *

In this theorem,

Given a lamination or a partial marking μ and subsurface *Y*, we say that μ and *Y* overlap, writing *Y*, we have *Y* and *Z*, if *Y* and *Z* overlap, and write

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

*There is a constant *

*whenever *

As shown in [26], the constant

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

*Given *

*i*. Then

### 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*, *f* is an orientation preserving homeomorphism to a finite-type Riemann surface *X*, where *X* as a point in Teichmüller space, with the equivalence class of marking implicit. We equip

Let *X* be a finite-type Riemann surface and let *X*. A quadratic differential *q* is a nonzero, integrable, holomorphic section of the bundle *q* has the form *w*, *q* changes by the square of the derivative, and is thus given by *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 *a natural coordinate**q* on *X*. Integrating *S* with singularities at the zeros. Using the identification *q*.

Now given a quadratic differential *q* on *X*, the associated Teichmüller geodesic is determined by the family of Riemann surfaces *q* at *X* and *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 *X*. Again abusing notation and denoting the point in *X*, we write the hyperbolic length simply as

The Thurston compactification,

for all simple closed curves

### 2.7 Some hyperbolic geometry

Here we list a few important hyperbolic geometry estimates. For a hyperbolic metric *width of α in X*. This is the width of a maximal embedded tubular neighborhood of α in the hyperbolic metric

*X*– that is,

*w*so that the open

*For any simple closed curve α, we have*

*Consequently,*

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

### 2.8 Short markings

For *L*-bounded length marking at *L*-short marking) is a marking with the property that any curve in *L*, and so that for each *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

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

where

### 3.1 Motivating example

The motivating examples are sequences of curves in

Now define

The first five curves,

Observe that for any

Furthermore, the next curve in the sequence,

### 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 *k*), where *G* is the constant from Theorem 2.11 and *E* is the constant in Theorem 4.1 below. For the examples in

In the next definition,

Suppose that *S* satisfies

- (i)
are pair-wise disjoint and distinct,${\gamma}_{k},\dots ,{\gamma}_{k+m-1}$ - (ii)
fill the surface${\gamma}_{k},\dots ,{\gamma}_{k+2m-1}$ *S*, - (iii)
, where${\gamma}_{k+m}={\mathcal{D}}_{{\gamma}_{k}}^{{e}_{k}}\left({\gamma}_{k+m}^{\prime}\right)$ is a curve such that${\gamma}_{k+m}^{\prime}$ (here we ignore any situation with$i({\gamma}_{k+m}^{\prime},{\gamma}_{j})\{\begin{array}{cc}\in [{b}_{1},{b}_{2}]& \text{for}j\in \{k-m,\dots ,k-1\},\\ =b& \text{for}j=k,\\ =0& \text{for}j\in \{k+1,\dots ,k+m-1\},\end{array}$ ).$j<0$

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 *b* constants, and the conditions on *a*.

Here we verify that the sequence of curves on

Since *k*.

Finally, note that *k* follows from this figure as well, after applying

Since *k* by inspection of Figure 1.

Returning to the general case, we elaborate a bit on the properties in

*For every *

*moreover *

Since

proving the first statement. For the special case

### 3.3 Visualizing the conditions of $\mathcal{P}$

The conditions imposed in *m* and *m* congruence conditions. It is useful to view the tail of the sequence starting at any curve

From the first condition of *b* times and it intersects everything in the row directly below it between *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

## 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

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.

*Given a surface S and *

- (1)
*the multi-curves*$\partial {Y}_{k},\dots ,\partial {Y}_{k+m-1}$ *are pairwise disjoint,* - (2)
${Y}_{k}\u22d4{Y}_{j}$ *for all* ,$j\in \{k+m,\dots ,k+2m-1\}$ - (3)
${d}_{{Y}_{k}}(\partial {Y}_{j},\partial {Y}_{{j}^{\prime}})>E$ *for any* ,$j\in \{k+m,\dots ,k+2m-1\}$ .${j}^{\prime}\in \{k-2m+1,\dots ,k-m\}$

*Then for every *

*and*

*Furthermore, suppose that for some *

- (4)
*the multi-curves*$\partial {Y}_{k},\dots ,\partial {Y}_{k+2n-1}$ *fill**S*.

*Then for any two indices *

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

Set the constants

Here *G* is the constant from Theorem 2.11 (Bounded geodesic image theorem) for a geodesic (i.e.

For the base of induction, suppose that

Similarly,

which is the bound (4.2).

Suppose that (4.1) and (4.2) hold for all

From the base of induction we already have

and

Consequently,

We now turn to the proof of (4.2). Since

Since

By Theorem 2.9,

Combining these two inequalities with (4.4), we obtain

This completes the first half of the double induction.

We now know that statements (4.1) and (4.2) hold for all

We may again write the triangle inequality (4.4). Since

and so Theorem 2.9 again implies

and Theorem 2.9 once again implies

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

Suppose that δ is any multi-curve. Let

*Suppose that *

Observe that by the claim,

Without loss of generality, we assume that *S*, and so there exists *t* with

Now observe that

On the other hand, since

a contradiction. ∎

Let η be a geodesic in

Thus Theorem 2.11 guarantees that there is a curve

Therefore, η contains at least

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

*Let *

By Theorem 4.1, inequality (4.3), the sequence

We complete this section by showing that

*All curves in a sequence *

Condition (i) of *j* and

### 4.1 Subsurface coefficient bounds

We will need estimates on all subsurface coefficients for a sequence satisfying

*Given a sequence *

- (1)
*If*$i,j,k$ *satisfy*$j\le i-m$ *and*$i+m\le k$ *, then* ,${\gamma}_{i}\u22d4{\gamma}_{k}$ ${\gamma}_{i}\u22d4{\gamma}_{j}$ *, and*${d}_{{\gamma}_{i}}({\gamma}_{j},{\gamma}_{k}){\stackrel{+}{\asymp}}_{R}{e}_{i}\hspace{1em}\text{\mathit{a}\mathit{n}\mathit{d}}\hspace{1em}{d}_{{\gamma}_{i}}({\gamma}_{j},\nu ){\stackrel{+}{\asymp}}_{R}{e}_{i}.$ - (2)
*If*$W\u228aS$ *is a proper subsurface,*$W\ne {\gamma}_{i}$ *for any**i**, then for any*$j,k$ *with*${\gamma}_{j}\u22d4W$ *and* ,${\gamma}_{k}\u22d4W$ ${d}_{W}({\gamma}_{j},{\gamma}_{k})<R\hspace{1em}\text{\mathit{a}\mathit{n}\mathit{d}}\hspace{1em}{d}_{W}({\gamma}_{j},\nu )<R.$

*Let μ be a marking on S. Then there is a constant *

*•**For any**k**sufficiently large and*$i\le k-m$ *we have*${d}_{{\gamma}_{i}}(\mu ,{\gamma}_{k}){\stackrel{+}{\asymp}}_{R\left(\mu \right)}{e}_{i}\hspace{1em}\text{\mathit{a}\mathit{n}\mathit{d}}\hspace{1em}{d}_{{\gamma}_{i}}(\mu ,\nu ){\stackrel{+}{\asymp}}_{R\left(\mu \right)}{e}_{i}.$ *•**For any proper subsurface*$W\ne {\gamma}_{i}$ *for any**i**we have*${d}_{W}(\mu ,{\gamma}_{k})<R\left(\mu \right)\hspace{1em}\text{\mathit{a}\mathit{n}\mathit{d}}\hspace{1em}{d}_{W}(\mu ,\nu )<R\left(\mu \right).$

We begin with the proofs of (4.5) and (4.6).
First note that since any accumulation point of *k* we have *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

Note that *G* depends only on *m*.
Similar reasoning implies

Combining these, we have

It follows that

We now move on to the inequalities in (4.6), and without loss of generality assume that

Next, suppose that

There are two cases to consider depending on whether

It follows from the previous paragraph that

hence

Now suppose that

Suppose then that *W* is not an annulus, then *W*. Therefore

since *W* is an annulus, because

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

Now we suppose that

Note that *j* and *k*, respectively (if *r* and *s*, it follows that

Note that since any *S*, either

have the desired properties. Indeed, *S*, there must be some

By the triangle inequality and (4.9) we have

Since

So the inequality on the left of (4.6) holds for any

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

## 5 Measures supported on laminations

In this section we begin by proving intersection number estimates for a sequence of curves satisfying *m* subsequences and prove that these converge in *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 *b* and sequence

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

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

Then *k* is in the first or second row. If *k* is below these rows, then the product defining *j* directly above *k*, up to and including the entry in the second row.

We now state the main estimate on intersection numbers.

*Suppose that ${\left\{{\gamma}_{k}\right\}}_{k=0}^{\infty}$ is a sequence on a surface S satisfying $P\left(a\right)$.
For a is sufficiently large, there is a constant $\kappa =\kappa \left(a\right)>1$, so that for each $i,k$ with $k\ge i+m$ we have*

Recall that for

Throughout all that follows, we will assume that a sequence of curves

### 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

By a second induction, we will bound *a*. Next, we will recursively define a function

For *a* sufficiently large, we prove

### Upper bound

Recall from

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)):

*For all *

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].

*Let δ, *

*e*

As above,

*For any curve δ and any *

Since

Assume all curves intersect minimally transversely and that there are no triple points of intersection. From the definition of *B*, all complementary components of *B* pairwise disjoint arcs of

Combining this with the above inequality proves the proposition. ∎

Fix *i*. The proof is by induction on *k*. For

so the lemma follows. Similarly, for

Applying Proposition 5.4 to the case

Therefore, we have

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

as required. ∎

Next we prove that

*There exists *

*a*,

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

*For all *

If

Now assume that

(where

terms in the product, indexed by

terms, indexed by

as required. ∎

As an application, of Lemma 5.6, we prove

*For all *

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

The proof is by induction on *k*. Since *k*. Now assume that *k* and at least *i*. Let

From this, the definition of

Since *k*. Indeed, the required bound for

This completes the proof. ∎

The upper bound on

where we have substituted *a*.

The infinite product converges if and only if the infinite series obtained by taking logarithms does. Since

The last expression is essentially a geometric series, and hence converges for all

### Lower bound

Let

(which is possible since

*For all *

Fix an integer *k*. For the base case, we let

Applying Proposition 5.4 to the curve

as required. ∎

*Set *

If *s* and *p* are positive integers with *m*. Note that

By Lemma 5.6, it follows that for all

Iterating this inequality *s* times implies

Since

This completes the proof. ∎

For

Since

### Convention

From this point forward, we will assume that

### 5.2 Convergence in $\mathcal{M}\mathcal{L}\left(S\right)$

Consider again a sequence of curves *m* convergent subsequences in

For each

where *A* is defined in (5.1).

For each

The main result of this section is the following theorem.

*Suppose that *

*in *

We will need the following generalization of Theorem 5.1.

*For any curve δ, there exists *

Note that in Theorem 5.1, we estimate *i*, and not *k*, so the lemma for

First we note that by Theorem 5.1, we have

From the definition of *A*, and the fact that

Setting

Next, let *S*, the set of measured laminations

is compact. From (5.7) we have

Let *i*, there exists

Let *i* again, we can take

Since every accumulation point of

for all

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

*For any curve δ and any *

By Proposition 5.4 we have

Dividing both sides by *i* sufficiently large

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

Since

Consequently, for all

By taking *i* and *j* sufficiently large, the (partial) sum of the geometric series on the right can be made arbitrarily small. In particular,

Fix

## 6 Ergodic measures

We continue to assume throughout the rest of this section that *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

The aim of this section is to show that

Using the estimates on the intersection numbers from Theorem 5.1, we first show that the measures

*Let *

*In particular, the measures *

The last statement is a consequence of the two limits, for if

For

Dividing the first equation by the second and taking limit (and doing the same with the roles of *h* and

To treat the two estimates in (6.1) simultaneously, we suppose for the time being that

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

We will simplify the expression on the right, but the precise formula depends on whether

where

Now observe that when

proving the first of the two required equations. So, suppose *i* terms in the product is bounded above by

where when *i* tend to infinity, we arrive at the second of our required estimates, and have thus completed the proof.
∎

We immediately obtain the following:

*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.

*If *

Recall that *d*-dimensional) space of measures on ν. For each

where *i*, *h*, and

Next, fix *h* and let

Now suppose that for some

This contradiction shows that

*If *

Since the dimension of the space of ergodic measures *d* is at most

### 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.

*Suppose that *

*m*subsequences with

Let

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 *S* from a marked Riemann surface *q* on *X* with at most simple poles at the punctures, so that the vertical foliation *q*-metric is computed by integrating

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

where

where *g* is divergent in the moduli space.
The vertical and horizontal measure of a curve γ is denoted

From this it follows that the natural area measure from *q*. Likewise, this area naturally decomposes as the push forward of the measures *X* and its image in *X* or

Given *-**thick subsurface* of *Y* and a continuous map *Y* with the following properties.

- (1)The boundary of
*Y*is sent to a union of -geodesics, each with extremal length less than$q\left(t\right)$ in${\u03f5}^{\prime}$ .$X\left(t\right)$ - (2)If
*Y*is not an annulus, then every nonperipheral curve in*Y*has -length at least ϵ and$q\left(t\right)$ *Y*has no peripheral Euclidean cylinders. - (3)If
*Y*is an annulus, then it is a maximal Euclidean cylinder.

We will be interested in the case that *Y*, which does not contain a Euclidean cylinder (i.e. a large modulus expanding annulus; see [36]). Consequently,

As an abuse of notation, we will write *Y* is only embedded on its interior. An *-**decomposition of * is a union of

with pairwise disjoint interiors. We note that

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

*With the assumptions on the Teichmüller geodesic g above, there exist constants *

- (1)
${Y}_{0}\left({t}_{k}\right),\dots ,{Y}_{d-1}\left({t}_{k}\right)$ *is an*$({\u03f5}_{k},\u03f5)$ *-**thick decomposition,* - (2)
${\mathrm{Area}}_{j}\left({Y}_{j}^{0}\left({t}_{k}\right)\right)>B$ *for all*$0\le j\le d-1$ *and for any component* ,${Y}_{j}^{0}\left({t}_{k}\right)\subset {Y}_{j}\left({t}_{k}\right)$ - (3)
${\mathrm{Area}}_{j}\left({Y}_{i}\left({t}_{k}\right)\right)<{\u03f5}_{k}$ *for all*$0\le i,j\le d-1$ *with* ,$i\ne j$ - (4)
.$\mathrm{Area}(X\left({t}_{k}\right)-({Y}_{0}\left({t}_{k}\right)\cup \dots \cup {Y}_{d-1}\left({t}_{k}\right))<{\u03f5}_{k}$

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

To apply this construction, we will need the following lemma. First, for a curve γ and

*Given any sequence *

*k*sufficiently large, each

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

Let *k* be large enough so that the conclusion of Lemma 6.8 holds. For each *m*, and consequently *m* curves. That is,

### 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

be the Teichmüller geodesic described above with vertical foliation

Suppose that *Y* is itself a Euclidean cylinder (in which case we assume it is maximal). As in the case of thick subsurfaces, we write *Y* is thick. Suppose that *Y* in arcs. This is the case for *k* sufficiently large, as well as any *Y* for which *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*

and *vertical strips*

Each horizontal strip

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

The *width* of a horizontal strip

*Let *

*is a decomposition into maximal horizontal and vertical strips, then*

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

To see this, we note that the area of *Y* is the sum of the areas of the horizontal (or vertical) strips. Every time *except*, near the ends of

If *Y* is nonannular, then note that

To see this, we note that the horizontal foliation (for example) is

Now suppose that

In particular, if

The balance time of γ along the Teichmüller geodesic *g* is the unique

Consider *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

We note that this estimate was under the assumption that

Suppose that *Y* is itself a Euclidean cylinder in which case we assume it is a maximal Euclidean cylinder. We further assume that

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

Combining this with (6.5) proves

The annular case is similar: Again by Proposition 4.5 since the core curve

## 7 Constructions

In this section we provide examples of sequences of curves satisfying

### 7.1 Basic setup

Consider a surface *S* and *m* pairwise disjoint, nonisotopic curves *k*, let *S* cut along *k* we assume the following:

- (1)
contains both$\partial {X}_{k}$ and${\gamma}_{k+1}$ (with indices taken modulo${\gamma}_{k-1}$ *m*), - (2)we have chosen
a${f}_{k}:S\to S$ *fixed*homeomorphism which is the identity on , and pseudo-Anosov on$S\setminus {X}_{k}$ ,${X}_{k}$ - (3)the composition of
and the Dehn twist${f}_{k}$ , denoted${\mathcal{D}}_{{\gamma}_{k}}^{r}$ , has translation distance at least 16 on the arc and curve graph${\mathcal{D}}_{{\gamma}_{k}}^{r}{f}_{k}$ for any$\mathcal{A}C\left({X}_{k}\right)$ ,$r\in \mathbb{Z}$ - (4)there is some
so that$b>0$ , independent of$i({\gamma}_{k},{f}_{k}\left({\gamma}_{k}\right))=b$ *k*.

For *k* to *h*, mod *m*. This means that if *k* to *h*, while if

If

For any

If *S* cut along *m*.

We also define

where we are composing

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

*For each *

In the case *m*), we note that

fills *S*. We also observe that for all

In the following proof, we write

The last statement follows from the first assertion since, for all

The conditions on the curves and homeomorphisms are symmetric under cyclic permutation of the indices, so it suffices to prove the lemma for *j*.

The base case is

Suppose that for some

Note that since

Now suppose that

a contradiction.

Therefore, δ is contained in

Now since δ is essential in

The last equality follows from the fact that

contradicting our choice of δ.
Therefore,

*For all *

We recall from the previous proof that *S*, but satisfies

Since

Since

Since

### 7.2 General construction

Let

For *m*, and for

The sequence of curves

- (1)The first
*m*curves are , as above.${\gamma}_{0},\dots ,{\gamma}_{m-1}$ - (2)For
, set$k\ge m$ ${\gamma}_{k}={\varphi}_{m}{\varphi}_{m+1}\cdots {\varphi}_{k}\left({\gamma}_{\overline{k}}\right).$

We could have avoided having the first *m* curves as special cases and alternatively defined a sequence *m* curves.

*With the conditions above, the sequence *

*b*is the constant assumed from the start).

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

*For any *

*(in the same order).
Furthermore, the homeomorphism can be chosen to take *

We prove the lemma assuming

Let

From these facts we observe that for

while for

This completes the proof of the first statement.

Next, since

Applying

Let *m* to *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

To prove that condition (iii) is also satisfied, we need to define

and verify the intersection conditions.
We fix

(the case of general

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

Therefore,

so

To prove the intersection number conditions on

The second-to-last equality is obtained by applying

On the other hand, for

by assumption (4).

Finally, similar calculations show that for

There are only finitely many possible choices of

While any sequence of curves as above satisfies the conditions in sections in

*Suppose that the sequence *

*S*.

First assume

For each *j*, *j* does not fall into one of the above two cases, then

When *S*. Consequently, *S*.

Now we must prove that for *S*. The proof is by induction, but we need a little more information in the induction. For simplicity, we assume that

To describe the additional conditions, for

With this notation, we now wish to prove by double induction (on *k* and

The base case is *S*. We note that applying

For the first and last curves

which has

But notice that

as required for the base case.

For the induction step, the proof is quite similar. We assume that the statement holds for all *S* and that

Therefore, applying

The homeomorphism

Since

Applying

In particular, we have

This proves part of the requirement on

We must also show that *S*. We will show that the *S*, which will suffice. To see this, take any essential curve δ and suppose it is disjoint from both

Since *S*, δ must intersect one of these curves. However,

This contradicts our initial assumption on δ, hence no such δ exists and *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 *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

### 7.3.1 Maximal-dimensional simplices

For the first family of examples, we can choose a pants decomposition on

### 7.3.2 Non-maximal examples

For our second family, we choose

## 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 *a*. For

Let ν be the nonuniquely ergodic lamination determined by the sequence (see Theorem 4.3 and Corollary 6.2). Furthermore let *h*. Let

for any

Let *X*. By [18], there is a unique Teichmüller geodesic ray starting at *X* with vertical foliation

We write

We also recall that

For any curve α let *active interval of *

Write *t* when

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

- (i)
*For**k**sufficiently large,*${J}_{k}\ne \varnothing $ *. Moreover,*${J}_{k}\cap {J}_{l}=\varnothing $ *whenever* .$i({\gamma}_{k},{\gamma}_{l})\ne 0$ - (ii)
*For*$0\le f<k$ *sufficiently large with* ,$k-f\ge m$ ${J}_{f}$ *occurs before*${J}_{k}$ *. Consequently, some tail of each subsequence*${\left\{{J}_{i}^{h}\right\}}_{i=0}^{\infty}$ *appears in order.* - (iii)
*For**k**sufficiently large and a multiplicative constant depending only on*ν*and**X*,${\mathrm{Hyp}}_{{a}_{k}}\left({\gamma}_{k}\right)\stackrel{*}{\asymp}\frac{1}{{d}_{{\gamma}_{k}}(\mu ,\nu )}\stackrel{*}{\asymp}\frac{1}{{e}_{k}}.$ - (iv)
*For an additive constant depending only on*ν,*X**, and**M**, we have*$\left|{J}_{k}\right|\stackrel{+}{\asymp}\mathrm{log}{d}_{{\gamma}_{k}}(\mu ,\nu )\stackrel{+}{\asymp}\mathrm{log}\left({e}_{k}\right).$

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

*With notation and assumptions above, there exists *

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

Let *g* be a geodesic in *S*, and *g* tends to infinity with *k*. For *Y* and *k* sufficiently large, *g*. Consequently, *g*, and Theorem 2.11 guarantees that *g* limits to η (or one of it is curves is disjoint from η), it follows that

From [36], if *M*. For all *k* sufficiently large, (4.7) and Lemma 8.2 imply

By construction, *k*.
Furthermore, for all

By (4.5) in Proposition 4.5 we have for all

Let

Since

for all

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

For *k* is sufficiently large, Lemma 8.2 implies

Taking logarithms we obtain

We proceed to the proof of part (iii). Following Rafi in [36, Section 6], we introduce the following constants associated to a curve *Y* is an annulus, recall that *Y*).

- •If
*Y*is a nonannular subsurface, an arc β in*Y*is a*common*if β transversely intersects α and*K*-quasi-parallel of and${\pi}_{Y}\left(\eta \right)$ for α and${\pi}_{Y}\left(\nu \right)$ *Y*Here$\mathrm{max}\{i(\beta ,{\pi}_{Y}\left(\eta \right)),i(\beta ,{\pi}_{Y}\left(\nu \right))\}\le K.$ denotes the arc-and-curve projection of η: the union of arcs and curves obtained by intersecting η with${\pi}_{Y}\left(\eta \right)$ *Y*(likewise for ν). Define , where$K\left(Y\right)=\mathrm{log}K$ *K*is the smallest number so that η and ν have a common*K*-quasi-parallel. - •If
*Y*is an annular subsurface, let .$K\left(Y\right)={d}_{Y}(\eta ,\nu )$

Now define

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

In what follows we show that for all sufficiently large *k*, *Y* with

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

thus *Y* is a nonannular subsurface with *k*,

If *Y* contains no curves

Then choosing the threshold *A* in Theorem 2.7 larger than the upper bound on these projections, and applying the theorem to

In this case we have *k*.

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

where *Y*. From (4.1) in Theorem 4.1 we see that

and

Choose the threshold constant *A* from Theorem 2.7 larger than the constant

Therefore, β is a *K*-quasi-parallel with

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

Next we list some estimates for the locations of the intervals

Let *i* sufficiently large

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

*For any *

*The additive error depends on X, *

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 *X* is a fixed surface and μ a short marking, we have

Since *t*, and *i* sufficiently large,

Since μ is a fixed set of curves and *i* sufficiently large. Thus from (6.1), for

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

Combining this with (8.5), we have

Solving for

*For any k sufficiently large, we have *

Let

Therefore

Let

Let *i* sufficiently large and

*For *

The proof is similar to the proof of [22, Lemma 7.3]. For the first inequality, note that

We show that

Now since

Next suppose that (8.7) holds so

Now since

To obtain a greater control over the arrangement of intervals

Such sequences exist simply by setting

Condition (8.8) has the following consequence.

*Suppose that a sequence *

Let

Moreover, by (8.8) we have

that is,

Thus (i) follows.

Now suppose that (8.7) holds so

where the second inequality holds because

*Suppose that the growth condition (8.8) holds. Then for *

Let

First suppose that (8.6) holds so

where the sequence tends to infinity as

Now suppose that (8.7) holds so

where again the convergence to infinity as

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

*For *

*Furthermore,*

## 9 Limit sets of Teichmüller geodesics

In this section, we continue with the assumptions from the previous section on the sequences *X*, and active intervals

In addition, we will need one more condition on

The additional condition is

- $\mathcal{P}$ (iv)Let α be any essential curve in
. Then there is no subsurface$S\backslash {\sigma}_{k}$ with$Y\subseteq S$ which is filled by a collection of the curves in the sequence$\alpha \subseteq \partial Y$ .${\left\{{\gamma}_{k}\right\}}_{k=0}^{\infty}$

Recall that when *Y* is an annular subsurface by *Y*.

Note that when *S*, condition $\mathcal{P}$ (iv) holds vacuously because there are no essential curves in

Under these assumptions, Theorem 1.4 from the introduction, which describes the limit set of

*The accumulation set of *

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 *k* sufficiently large. Combining this with Lemma 8.8, it follows that for all *l*. Therefore, the set of intervals *k*, cover all but a compact subset of

*Fix *

*i*. Then

*Furthermore, if *

From the second part of Theorem 9.3 applied to