Let be a -diffeomorphism with Axiom A and no cycle condition on a two-dimensional smooth manifold. In this article, we prove that if is -robustly weak measure expansive, then it is -Anosov. Moreover, we expand the results of the -diffeomorphism case into the -vector field on a three-dimensional smooth manifold. Let be a -vector field with Axiom A and no cycle condition. We prove that if is -robustly weak measure expansive, then it is -Anosov.
Under mainly -topology, the main research topic of dynamical systems is the study of hyperbolicity using various properties for diffeomorphisms or flows. Franks’ lemma  and Closing lemma , famous properties in dynamical system studies, only work well for the -topology and play an essential role several times in the proof. In fact, we can see that Franks’ lemma is used in the proof process of [3,4] and , and the closing lemma is used in  and . To study more similar (closer) dynamics to a given dynamics, many mathematicians are studying -topology. The papers that directly motivated our research are the following.
(Pujals and Sambarino ) Let be a surface diffeomorphism and be a compact -invariant set admitting a dominated splitting Assume that all periodic points in are hyperbolic saddles. Then where
is a hyperbolic set and
is the union of finitely many pairwise disjoint normally hyperbolic circles
From this result, the following two theorems have been proved by adding a shadowing property view point.
(Sakai ) Let satisfy -stably shadowing property on a compact surface. If periodic points of are dense in the non-wandering set and there is a dominated splitting on the closure of periodic points of saddle type, then satisfies both Axiom A and the strong transversality condition.
(Lee ) Let satisfy -stably inverse shadowing property on a compact surface. If periodic points of are dense in the non-wandering set and there is a dominated splitting on the closure of periodic points of saddle type, then satisfies both Axiom A and the strong transversality condition.
In addition, Artigue showed the following by adding an expansivity view point in .
(Artigue ) Let be a -diffeomorphism with Axiom and no cycle condition on a compact surface. If it is -robustly CW-expansive, then is -Anosov. Here, the meaning of a -Anosov diffeomorphism is that it is Axiom A, has no cycles, and there is no 2-tangency.
It is well known that the relationship between the various expansivities is as follows.
In particular, we refer to  for an example of a homeomorphism that satisfies weak measure expansivity but not measure expansivity. More precisely, an irrational rotation map on the unit circle is weak measure expansive but not Lebesgue measure expansive. From this example, we can see that weak measure expansivity is clearly different from other expansivities, and it can be seen that the research value is a sufficient subject.
Theorem D became the motivation of one of the main theorems in this article as follows.
Let be a -star diffeomorphism with Axiom A and no cycle condition on a two-dimensional smooth manifold. If f is a -robustly weak measure expansive diffeomorphism, then it is -Anosov.
This problem is worth studying because there is no relationship between CW-expansivity and weak measure expansivity. Particularly, -Anosov is quasi-Anosov and quasi-Anosov is closely related to expansivities. The fact that a -robustly expansive, -expansive, CW-expansive, measure expansive, and weak measure expansive diffeomorphism is quasi-Anosov has already been proven. However, since we do not know the relation between weak measure expansive diffeomorphism and -Anosov in -dynamics, we will prove it in Theorem 1.1.
Moreover, we propose an extension of the -Anosov definition of a diffeomorphism to a flow and prove the following second main result.
Let be a -vector field with Axiom A and no cycle condition on a three-dimensional smooth manifold. If X is a -robustly weak measure expansive vector field, then it is -Anosov.
This result extends the result of the first main theorem, which is obtained for the case of diffeomorphisms to the case of continuous flows.
2 Basic definitions
2.1 Discrete dynamics
Let be a compact smooth -dimensional manifold without boundary and let ( ) be a set of diffeomorphisms with the topology. Let be the Borel -algebra on Denote by the set of Borel probability measures on endowed with the weak topology. We say that is atomic if there exists a point such that Let be the set of nonatomic measures Recently, Morales and Sirvent  introduced a general notion of expansivity as follows: Let be given, we say that is -expansive if there exists a constant such that for all where Such a is called an -expansivity constant of f.
From this, a concept of weak measure expansivity is introduced in . It is the generalizing notion of measure expansivity and it is based on the concept of measure-sensitive partition in . To do this, we say that a finite collection of subsets of is a finite -partition ( ) of if
’s are disjoint, and ;
each is measurable for a Borel probability measure, , and for all
 For any a homeomorphism is said to be weak -expansive if there is a finite partition of such that for all where
The set is called the dynamic -ball of with respect to , and denotes the element of containing . Denote by for simplicity if there is no confusion. Note that
A diffeomorphism is called weak measure expansive if it is weak -expansive for all
Note that if a diffeomorphism is weak -expansive for then is clearly nonatomic. Therefore, we can assume that is always an element of in this article.
Given we can take , and a -coordinate chart , (here is a neighborhood of ) such that and two functions such that the graph of and is the local expression of the local stable and the local unstable manifold of respectively. If the degree Taylor polynomials of and at 0 coincide we say that there is an -tangency at
 We say that -diffeomorphism is -Anosov if it is Axiom A, has no cycles, and there is no -tangency.
Particularly, -Anosov is quasi-Anosov and quasi-Anosov is closely related to expansivities. It has already been demonstrated that a -robustly expansive, -expansive, CW-expansive, measure expansive, and weak measure expansive diffeomorphism is quasi-Anosov. However, since we do not know the relation between weak measure expansive diffeomorphism and -Anosov in -dynamics, we will prove it in Theorem 1.1.
2.2 Continuous dynamics
Let ( ) be the set of vector fields endowed with the topology.
Then every generates a flow that is, a family of diffeomorphisms on such that for all , is the identity map and for any Here is called the integrated flow of .
For each the map defined by is a diffeomorphism. In addition, and we denote the orbit of x with respect to as
For any subset we say if for every Borel set Let be the set of Borel probability measures on endowed with the weak topology and
For any , we say that is -expansive if there exists a constant such that for all where for some and all Here is the set of continuous maps with and is said a reparametrization. Such a is called an -expansivity constant of f.
For a finite partition of and the dynamic -ball of with respect to is defined by
where stands for the element of containing . Denote by for simplicity if there is no confusion.
 For any , is said to be weak -expansive if there is a finite partition of such that for all We say that is weak measure expansive if it is weak -expansive for all .
3 Partially hyperbolic diffeomorphisms
In this section, we assume that is a compact surface, i.e.,
[11, Lemma 4.1] Let and be a neighborhood of . Then there are and an one-parameter family of diffeomorphism such that for all for all Moreover, the function is continuous in the -topology.
Recall that a point is called a (Lyapunov) stable point for if for any there is such that if then for all Since is compact, we can take subsequences and converging to points and , respectively, such that and for every For we say that is periodic of period for if for some but for all We denote by the period and by the set of periodic points of A point is called nonwandering of if for any neighborhood of in there is such that The set of nonwandering points of is called the nonwandering set of and is denoted by It is clear that
If f is a weak measure expansive diffeomorphism on M then no periodic point is stable.
Assume that admits a stable point Let be a finite -partition of for any Then there exists such that
for some where is the closed -ball centered at Since is a stable point, as For sufficiently large
Put and let be a normalized Lebesgue measure on Define by
for any measurable set of (this is well-defined). Then we obtain by (3.1) and this is a contradiction. So proof is completed.□
For a diffeomorphism , a property “P” is said to be -robust if there is a -neighborhood of such that for any satisfies “P”.
Let be a diffeomorphism on and be a periodic point of We say that is sink (resp. source) if all eigenvalues of have norm less than 1 (resp. bigger than 1). In other cases, we call is saddle.
If is -robustly weak measure expansive diffeomorphism on , then every periodic point of with period is saddle.
Let be a neighborhood of such that for all Suppose that the eigenvalues of are smaller or equal than 1 in modulus. By Lemma 3.1, take a -diffeomorphism of fixing and being the identity outside In particular, is the identity in a neighborhood of the points Assume that is closer to Define Then
is a periodic point of with period
is -close to , and
eigenvalues of are (where : eigenvalues of ) such that modulus (strictly) smaller than
Finally, if the eigenvalues of are bigger, or equal to 1 in modulus, then we can obtain the contradiction through a similar method to the above process. Therefore, we can complete the proof.□
We say that is a -star diffeomorphism if there is a -neighborhood of such that every periodic orbit of every is a hyperbolic set. In the case of , Smale et al. proved that the following three statements are equivalent.
satisfies Axiom A and no cycle condition,
is a star diffeomorphism.
However, we do not yet know whether an equivalence relation exists in -dynamics, so we propose the following question. “If a diffeomorphism of 2-dimensional manifold or any dimensional manifold is -star, then does it satisfy Axiom A?” Below we give a partial answer to the question above.
Every -robustly weak measure expansive diffeomorphism on a compact surface is a -star diffeomorphism.
To derive a contradiction, we suppose that there is a non-hyperbolic point for some Here, is a -neighborhood of and is the set of periodic points of Then has either only one eigenvalue with or only one pair of complex conjugated eigenvalues.
Case 1 : Let and (or ).
Case 2 : We can prove the second case similarly and complete the proof.□
The following proof is essentially contained in the proof of Theorem 4.4 in .
Proof of Theorem 1.1
Suppose that is not -Anosov. Then there is a wandering point with an 2-tangency. Take a -local coordinate at where are open sets. Let be functions which describe the graphs of the local stable and local unstable manifolds of respectively.
By the hypothesis, there exists a 2-tangency at So we may assume that the Taylor polynomials of order 2 of and vanish at 0.
Define the -diffeomorphism by
For define by
where is a -function such that
Let be defined by
Let If then Then we have
that is, is a map that sends the graph of to the graph of
Let be a finite -partition on There are and an arc such that
if is finite,
for some , and
for some containing
Let be a normalized Lebesgue measure on Define by
for any measurable set of (this is well-defined). For the wandering point the dynamic -ball of with respect to for all contains the arc Then we obtain
This fact shows that is not weak measure expansive, i.e., is not -robustly weak measure expansive. Therefore, this contradiction proves the theorem.□
4 Partially hyperbolic flows
Let be as before with Let be a closed orbit of a vector field Through a point we consider a section transversal to the field
The orbit through returns to intersection at time where is the period or By the continuity of the flow of the orbit through a point sufficiently close to also returns to intersection at a time close to Thus, if is a sufficiently small neighborhood of we can define a map which to each point associates the first point where the orbit of returns to intersection This map is called the Poincaré map associated with the orbit (and the section ).
For any and is singular if Denote by Sing the set of singular points of We say that is periodic if for some but for all and denoted by is the set of periodic point of And is regular if it is not singular nor periodic. A singularity or a periodic orbit of are both called a critical orbit or a critical point of and we denote by is the set of critical orbits.
The next lemma which is called tubular flow theorem describes the local behavior of the orbits in a neighborhood of a regular point.
 Let and let be a regular point of X. Let and let be the vector field on C defined by Then there exists a diffeomorphism , for some neighborhood of in taking orbits of to orbits of .
A tubular flow for is a pair where is an open set in and is a diffeomorphism of onto the cube and , which takes the orbits of in to the straight lines If denotes the field in induced by and that is, then is parallel to the constant field
The open set is called a flow box for the vector field By the tubular flow theorem, we know that if is a regular point of , then there is a flow box containing
Let be a vector field on , a closed orbit of , and a transversal section through a point Let be a neighborhood of and let be neighborhood of such that, for all the Poincaré map of is defined on
Next lemma is a -vector field version of [11, Lemma 4.1].
Let and . For any neighborhood of in and there are and a one-parameter family of -vector field such that
for any the flow is generated by the Poincaré map is defined by then one has for all
Let be a tubular flow with center such that is the constant vector field on Let be a vector field on such that is transversal to and and each orbit of through a point of meets Then we can define a map which associates with each point of the intersection of its orbit with By the Tubular Flow Theorem, is a diffeomorphism.
We claim that, given a neighborhood of and there is such that for any we can find a vector field such that on on , and if
Let be a function such that
Take such that
We define and find where satisfies the required conditions. We can immediately see from the definition that is established on . The differential equation associated with can be written as
Let satisfy We have and in a neighborhood of (it is clear by the definition of ). The solution of (4.1) with initial conditions and can be written as
By the continuity of there is such that for all i.e., on Then we obtain
on by solving the differential equation. Put . Then clearly and Take Then since is increasing, for we have Thus, Therefore, for all so that
It is straightforward to find out that we can select to form for so that Moreover, we explicitly obtain by taking small.
Let be the vector field on , which is equal to outside and on It is clear that is and The expression for the Poincaré map in the local chart is if Thus, for all □
Let X be a weak -expansive vector field for any . Then , where for any open neighborhood of
Since is weak expansive for every there is a finite partition of satisfying for all Let be a singularity of and It is clear that Thus, we obtain
this fact means □
We state that a point is called a stable point for a vector field if for any there is such that if then for all
If is a weak measure expansive vector field on then every closed orbit is not stable. We already proved this fact holds for diffeomorphisms in Lemma 3.2. This property can naturally be extended to flows.
For a vector field a property “P” is said to be -robust if there is a -neighborhood of in such that for any satisfies “P.”
A compact invariant set is transitive if for some and attracting if for some neighborhood of satisfying for all An attractor of is a transitive attracting set of and a repeller is an attractor for A sink of is a trivial attractor of namely it reduces to a single orbit of and a source of is a trivial repeller of Otherwise, the single orbit is called a saddle.
If is a -robustly weak measure expansive vector field on , then every periodic orbit is a saddle.
To induce contradiction, assume that there exists with period which is not saddle. This means that for a Poincaré map of there is with period such that has eigenvalues and (or and respectively). Let be a neighborhood of By Lemma 4.2, take a Poincaré map of fixing and being identity outside Here Assume that is closer to 1. Then
is a periodic point of with period
is -close to , and
eigenvalues of are such that modulus (strictly) smaller than 1.
We say that is a -star vector field if there is a -neighborhood of such that every critical orbit of every is a hyperbolic set. In the case of Gan and Wen proved that any nonsingular star flow satisfies Axiom and the no cycle condition in . However, as in the case of discrete dynamics, we do not yet know whether an equivalent relationship exists in -dynamics in continuous dynamical systems, so we propose the following question. “If a vector field on 3-dimensional manifold or any dimensional manifold is -star, then does it satisfy Axiom A?” In the remainder of the article, we give a partial answer to the aforementioned question.
Every -robustly weak measure expansive vector field X on M is a -star vector field.
To derive a contradiction, we suppose that there is a nonhyperbolic orbit for some Here, is a -neighborhood of Let and let be a Poincaré map of associated with the orbit (and the section ). Then has an eigenvalue with or only one pair of complex conjugated eigenvalues.
Case 1 : Let and (or ).
Then by Lemma 4.2, we can make a -vector field and Poincaré map such that eigenvalues and of are less than 1 (or bigger than 1). This means that has a sink point and this is a contradiction by Remark 4.4.
Case 2 : We can prove the second case similarly and complete the proof.□
We say that -vector field is a -Anosov vector field if it is Axiom A, has no cycles, and there is no 2-tangency.
Proof of Theorem 1.2
Suppose that is not -Anosov. Since is Axiom and has no cycle condition, there exists a 2-tangency According to Lemmas 4.1 and 4.3, we consider a normal section then there is a local chart and open sets such that Then it is clear and we let We can take a -local coordinate around where are open sets.
Let be functions such that their graphs describe the local stable and local unstable manifolds of in coordinates and let As in the proof of Theorem 1.1, we find maps and can define ( ) by
where is a time-1 map with respect to Let be a finite -partition on Then the same way of proof of Theorem 1.1, we can take an enough small constant and an arc such that
The authors wish to express their appreciation to Xiao Wen for his valuable comments.
Funding information: J. Ahn was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2020R1I1A1A01056614). M. Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Education) (No. NRF-2020R1F1A1A01051370).
Conflict of interest: The authors state no conflict of interest.
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