Measurable Versions of Some Topological Properties for Diffeomorphisms with Continuous Foliations

Wang, Lin; Wang, Xinsheng

doi:https://doi.org/10.1155/2023/6050079

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2023 | Article ID 6050079 | https://doi.org/10.1155/2023/6050079

Measurable Versions of Some Topological Properties for Diffeomorphisms with Continuous Foliations

Lin Wang¹and Xinsheng Wang²

Academic Editor: Lei Wang

Received13 Nov 2022

Revised01 Apr 2023

Accepted07 Apr 2023

Published21 Apr 2023

Abstract

In this paper, the concepts of plaque expansivity, topological quasi-stability, and quasi-shadowing property for Borel measures are considered. It is proved that every plaque expansive measure with the quasi-shadowing property is topologically quasi-stable with respect to its continuous foliation. At the same time, some other properties of the topological quasi-stable measures, the plaque expansive measures, and measures with the quasi-shadowing on a compact Riemannian manifold are investigated.

1. Introduction

It is well known that all of the expansivity property, shadowing property, and topological stability play important roles in the modern theory of dynamical systems, especially differentiable dynamical systems. The concept of expansivity for dynamical systems was first conceived by Utz [1] in the middle of the twenty century, which describes points of two distinct trajectories cannot be uniformly close to each other. It has been one of the main properties of hyperbolic dynamics generated by Anosov [2], quasi-Anosov [3], and pseudo-Anosov [4] diffeomorphisms on smooth compact manifolds. The shadowing property describes the behaviour of pseudo-orbits on or near a hyperbolic set. The classical shadowing lemma states that every pseudo-orbit lying in a small neighborhood of a hyperbolic set stays uniformly close to some true orbit (with slightly altered initial position). Roughly speaking, one says that a homeomorphism on a compact space is topologically stable if for any homeomorphism which is close to , there exists a continuous map from into itself such that and approaches to as approaches to , where is the identity map on . These three properties are not only important in their own right but also closely related to each other. In [5], Lewowicz demonstrated that expansivity is closely related to the topological stability by introducing some nonclassical Lyapunov functions. In [6], Walters showed that every expansive homeomorphism of a compact metric space with the shadowing property is topologically stable (see also [7]). Because the hyperbolicity implies the expansivity and the shadowing property, any Anosov diffeomorphism is topologically stable ([8]).

In [9], Morales and Sirvent introduced a notion of measure expansiveness which generalizes the usual concept of expansiveness. Several interesting properties of measure expansiveness have been obtained elsewhere [10–14]. In particular, Lee and Morales [14] introduced the counterpart concepts of topological stability and shadowing property of homeomorphisms for Borel measures. They got a measurable version of the classical result from Walters [6] that every expansive measure with the shadowing property is topologically stable.

We noticed that general nonhyperbolic diffeomorphisms may not have the expansivity property, for example, the identical mapping of a smooth manifold. The most common generalization of an Anosov diffeomorphism is the so-called partially hyperbolic diffeomorphisms. For partially hyperbolic diffeomorphisms, a center direction is allowed in addition to the hyperbolic directions. However, the center bundle is not always integrable [15]. If it is integrable, there exists a foliation of center manifolds usually denoted as . In [16, 17], the notions of center pseudo-orbits and plaque expansivity property with respect to the center foliation were introduced to investigate the robustness of the center foliation for normally hyperbolic and partially hyperbolic systems respectively. In [18], Kryzhevich and Tikhomirov gave a version of center shadowing property for dynamically coherent partially hyperbolic diffeomorphisms. At the same time, Hu et al. [19] showed that for any partially hyperbolic diffeomorphism , without any additional assumption, the quasi-shadowing property holds. In particular, if has center foliation, they can also obtain the similar result to that in [18]. In [20], Kryzhevich showed that the dynamical coherence implies the plaque expansivity for any partially hyperbolic diffeomorphism. As applications, Hu et al. [21] got the quasi-stability property by applying quasi-shadowing and plaque expansivity properties under the assumption of dynamical coherence. Wang and Zhu [22] used the quasi-shadowing property to get the relation between the growth rate of the number of periodic center leaves and the topological entropy. Later, Sun and Yang [23] obtained that any pseudo-orbit can be shadowed by a true orbit, and hence, they got that the growth rate of the number of hyperbolic periodic points is equal to the topological entropy for thin trapped chain hyperbolic homoclinic classes with one-dimensional center in the setting of partial hyperbolicity.

As we mentioned above, it is meaningful to investigate the quasi-shadowing property and the topological quasi-stability with respect to the center foliation for partially hyperbolic diffeomorphisms. We also noticed that, in [16], a continuous foliation can be defined and studied as a more general invariant manifold other than partially hyperbolic systems. For general theory of these invariant manifolds and recent progress, we refer to [16, 17, 24–29]. So we can specifically study the systems with quasi-shadowing and topological quasi-stability with respect to a continuous foliation, then partially hyperbolic systems with quasi-shadowing and topological quasi-stability with respect to the center foliation can be regarded as a special case.

As we all know, every expansive homeomorphism with the shadowing property of a compact metric space is topologically stable ([6]). However, because of the limitations of existing methods and techniques, quasi-shadowing and plaque expansivity cannot guarantee that quasi-stability holds for a diffeomorphism with a continuous foliation. In particular, even for a partially hyperbolic diffeomorphism, it is necessary to apply these two properties to get the topological quasi-stability with respect to center foliation under the assumption of dynamical coherence ([21]). Motivated by the concepts of expansiveness, topological stability, and shadowing property of homeomorphisms for Borel measures in [9, 14], we can investigate the quasi-shadowing property, topological quasi-stability, and plaque expansivity properties with respect to a continuous foliation for Borel measures and obtain a measurable version of those results as in [14]. Indeed, we introduce the notion of measure plaque expansiveness with respect to a continuous foliation, generalizing the usual concept of plaque expansiveness. We also introduce the concepts of topological quasi-stability and quasi-shadowing property for Borel measures and obtain that every plaque expansive measure having the quasi-shadowing property is topologically quasi-stable with respect to a continuous foliation. Certainly, the above conclusion is true for the smooth center foliation of a partially hyperbolic diffeomorphism ([16]). It should be stated that the main task of this paper is to study the measurable versions of topological properties for a diffeomorphism with a continuous foliation rather than a partially hyperbolic diffeomorphism with a smooth center foliation.

This paper is organized as follows. In Section 2, we give some basic definitions and notions used in this paper, including plaque expansive measures, topological quasi-stable measures, and measures with quasi-shadowing property. Finally, we discussed a number of general results on measures with the above properties in Section 3.

2. Basic Definitions and Notions

In this paper, we assume that is a smooth -dimensional compact Riemannian manifold. We denote by and the norm on and the metric on induced by the Riemannian metric, respectively.

First of all, let us consider the following definition of continuous foliations.

Definition 1. (see [16]). A continuous foliation of a manifold (with leaves of dimension ) is a disjoint decomposition of into -dimensional injectively immersed connected submanifolds which is called the leaves of the foliation such that is covered by charts, i.e., there exist the foliation coordinate charts and , where is the unit ball. The foliation is of class if the charts can be chosen of class .
By the above definition, a continuous foliation of a manifold is of class , and locally, as . For any , denote the submanifold (or leaf) of . For , denote the local submanifold of size . It is clear that is a closed subset of . Let be a closed subset of , if the foliation is continuous, then is also a closed subset of .
A foliation is of leaves ([16]) if its leaves are coherently immersed submanifolds. That is, for , the tangent planes of the leaves give a continuous -plane subbundle of . For , the -th order tangent multiplane to the leaf at depends continuously on .
The following definitions generalize some ideas, such as shadowing property, topological stability, and expansivity, of dynamical systems to invariant foliations.

Definition 2. We say that a diffeomorphism of has a continuous foliation if and only if it sends onto . We also say or is -invariant.
Firstly, let us recall the definition of the shadowing property.
Given , a sequence of points is said to be a -pseudo orbit for , ifGiven , we call that a -pseudo orbit is “-shadowed” (or “-traced”) by a trajectory of , where for some , if is said to have the shadowing property if for any , there exists such that for every -pseudo orbit , it can be -shadowed by some trajectory of .
An -pseudo orbit is called an --foliation pseudo-orbit if for any , the inclusion holds.

Definition 3. Suppose has a continuous foliation . We say that has the quasi-shadowing property with respect to if for any , there exists such that for any -pseudo orbit , there exists an --foliation pseudo orbit of -shadowing it, i.e., (2) holds.
Note that, for the shadowing property, any -foliation pseudo-orbit is a true trajectory.
Then, we recall the notion of topological quasi-stability property. For any maps defineDenote by the identity map. Denote is a homeomorphism and , and we call that the homeomorphism is -close to if .

Definition 4. Let be a diffeomorphism with a continuous foliation . We say that is topologically quasi-stable if for any , there exists such that for each homeomorphism that is -close to , there exist a continuous map and a family of motions which move points along the foliation and continuously depend on satisfying(1);(2).Recall that is expansive if there is (called expansiveness constant) such that for all implies , which can also be expressed as for all , here .

Definition 5. We say that is plaque expansive with respect to a continuous foliation if there exists (called plaque expansiveness constant) such that for any --foliation pseudo-orbits and , if for all , then for all .
DenoteTake the k-th position element from and denotethen the above definition is equivalent to for any --foliation pseudo-orbit . For simplicity of notation, the notation will be replaced by without causing confusion.

Remark 1. For a partially hyperbolic diffeomorphism with uniquely integrable center bundle (its maximal integral manifolds generate a center foliation of ), Hirsch et al. [16] proved that the plaque expansivity with respect to the center foliation not only guarantees the integrability of the center distribution for sufficiently small perturbations of but also survives under small perturbations. However, in the general case, a partially hyperbolic diffeomorphism with uniquely integrable center bundle cannot guarantee that plaque expansivity holds, so some other conditions need to be given. Hirsh et al. [16] got that if its center bundle is uniquely integrable and the corresponding center foliation is smooth or if acts as an isometry for every , then satisfies the plaque expansivity. Without assumption on smoothness of , Kryzhevich and Tikhomirov gave a conjecture that if is dynamically coherent, then satisfies the plaque expansivity.
Now, we proceed to generalize the notions of plaque expansivity, quasi-shadowing property, and topological quasi-stability for Borel measures. Recall that the Borel -algebra of is the -algebra generated by the open sets, whose elements can be called the Borelians of . A Borel measure defined on the Borelians of is a -additive measure. We assumed that any Borel measure is to be nontrivial, i.e., . Recall that a Borel measure is expansive with respect to if there is such that for any . Just as Morales and Sirvent [9] introduced the notion of expansive measures, the notion of plaque expansivity can also be defined for Borel measures as follows.

Definition 6. A Borel measure is plaque expansive with respect to the foliation of if there is (called plaque expansiveness constant) such that , for every --foliation pseudo-orbit .
Note that, we do not require that the measure in this definition be invariant namely for every Borel set . With these definitions, we can define the quasi-shadowing property for Borel measures. Fixed , recall that a sequence is through if .

Definition 7. A Borel measure has the quasi-shadowing property with respect to the foliation of if for every , there exist and a Borelian with such that every -pseudo orbit through can be -shadowed by an --foliation pseudo-orbit of .
In order to give topologically quasi-stable measure, we firstly define a motion along the foliation between two sets using some basic ideas which come from set-valued analysis ([30]).
Let , we say that belongs to the foliation -neighborhood of which is denoted by if for any , there exists such that . In fact, if , then for any , there exists and a motion along the foliation from to such that . So we can define that is an -motion along the foliation from subset to subset which is denoted by , if and . Particularly, if , we think that there still exists an -motion such that .
We represent the set formed by the subsets of as . A map will be referred to as a set-valued map of . We define the domain of by . We say that is compact-valued if is compact for every . We write for some if , where denotes the closed ball operation (note that such an inclusion is obviously holds for points . A set-valued map of is upper semicontinuous if for every and every neighborhood of , there is such that whenever satisfies .
By the definition of topologically quasi-stable diffeomorphisms, for a Borel measure , we give the following definition.

Definition 8. A Borel measure is called to be topologically quasi-stable with respect to the foliation of if for every , there is such that for every homeomorphism that is -close to , there is an upper semicontinuous compact-valued map of with measurable domain and a family of -motions which move sets along the foliation and are upper semicontinuously dependent on such that(1)(2)(3)(4)

Remark 2. In Definition 8, the family of -motions is upper semicontinuously dependent on means that if for every and every neighborhood of , there is such that whenever satisfies . Here, we write the motions in the form to indicate their dependence on . For brevity of expression in the following, we may use the expression “a Borel measure is topologically quasi-stable with respect to the foliation of ” rather than that “ with foliation is topologically quasi-stable with respect to a Borel measure .”

3. Main Results

In this section, we shall obtain some general results related to plaque expansive measures, topological quasi-stable measures, and measures with quasi-shadowing property.

3.1. Measures with Plaque Expansivity, Quasi-Stability, and Quasi-Shadowing Property

In this subsection, we will obtain some results which demonstrate that it makes sense to define plaque expansivity, topological quasi-stability, and quasi-shadowing property for Borel measures. For a partially hyperbolic diffeomorphism, Hu et al. [21] showed that once the center foliation exists and is dynamically coherent with the stable and unstable foliations, then the quasi-stability is obtained via quasi-shadowing property. Note that for any partially hyperbolic diffeomorphism, dynamic coherence implies continuity of the center foliation and plaque expansiveness with respect to the center foliation. We will first give a measurable result as in [14] and mainly prove the following result under the condition that has a continuous foliation. However, this result is on Borel measures rather than diffeomorphisms.

Theorem 1. Every plaque expansive measure with the quasi-shadowing property is topologically quasi-stable with respect to a continuous foliation of .

Proof. Let be a plaque expansive measure and have the quasi-shadowing property with respect to a continuous foliation of . Denote the plaque expansiveness constant of by . For any , we take . For this , let and be given by the definition of quasi-shadowing property of . Give a homeomorphism satisfying and we can define the set-valued map on as follows:It is easy to get that is a compact-valued map on .
Then, we show that is measurable. Choose a sequence converging to some . Because each , we can choose a sequence where such thatSince is compact, for , we can assume that for some . Through using the sieve method and the continuity of the foliation, we can find other () satisfying for , where for . Fix , we haveAnd letting , we get . Hence, we have and thus . Then, is closed and so we can get is measurable.
Next let’s show that . Because of , we can get the -orbit of each is a -pseudo orbit of . By choosing , we get a -orbit which is through . So can be -shadowed by an -W-foliation pseudo-orbit of . It follows that there exists a sequence where such that for every . From this, we have for every . Then, and so .
And then, we show that is upper semicontinuous. Let and be an open neighborhood of . DefinewhereBy the definition of , for any and , each is compact and . Hence, we can get such that . We declare that there is such that if . If not, there is a sequence satisfying and for each . By the definition of , for each , there exists the corresponding sequence with . Since is compact, by using the sieve method and the continuity of the foliation, we can get satisfying , where for . Clearly . However, and for and . Then, we haveLetting we have for . Hence, . Because of and , we can get a contradiction which shows the declaration. By the declaration, we have if . So is upper semicontinuous.
Now let’s prove , that is, for each . Choose and . If another point , we can get a sequence with such that for every . Since , we can also get a sequence with such that for every . All together imply for all . As , for , we conclude that thus proving . Therefore, as is an expansiveness constant of . The above shows that . By the definition of in (6), we get . Because of , we also obtain .
Finally, we prove that there is an -motion along the foliation such that . By the definition of in (2), we haveDenotethen we have and . If , and also . By Lemma 1 below, there exists an -motion along the foliation such that , i.e., . Then, is -invariant and in . For , we also have and so . Hence, we also think that there exists an -motion such that in . Since is an upper semicontinuous compact-valued map of , is continuous on and (4) of Definition 8 holds, we can get the upper semicontinuous dependence of on .

Lemma 1. Let and be two subsets of , if then , and there exists an -motion along the foliation such that .

Proof. For any , we have and . Since , there exists such that . Then, and . So we have . By the same ways as above, we also have . This shows that there exists an -motion along the foliation such that .
We say that is an atom of a Borel measure if and is nonatomic if it has no atoms. As far as we know, each nonatomic measure is expansive with respect to any expansive homeomorphism. Similarly, for topologically quasi-stable diffeomorphisms, we get the following theorem.

Theorem 2. Every nonatomic Borel measure is topologically quasi-stable with respect to any topologically quasi-stable diffeomorphism.

Proof. We assume that is a topologically quasi-stable diffeomorphism of a compact Riemannian manifold and is a nonatomic Borel measure of . Let and witness parameters in the definition of topological quasi-stability for . Choose a homeomorphism satisfying . Then, we can get a continuous map and a family of motions which move points along the foliation such that and . Next, we can define the set-valued map of as follows: for . It is easy to get that and is compact-valued. What’s more, because of the continuity of , we also get that is upper semicontinuous. As , we get . In addition, is nonatomic, so for each . Then, we have . We also have for all thus . Since , we have . This proves the result.

Remark 3. We have known that partially hyperbolic dynamically coherent diffeomorphisms on compact manifolds are topologically quasi-stable ([21]), so by the above theorem, we can get that, for all such diffeomorphisms, each nonatomic Borel measure is topologically quasi-stable.
We say that that a diffeomorphism has a compact foliation if the foliation of is a continuous and is compact for each . Recall that the compact foliation is uniformly compact ifwhere is the Riemannian volume restricted to the submanifold of .

Remark 4. If we assume that the partially hyperbolic diffeomorphism has a uniformly compact center foliation, then by Theorem 1 of [31] and Proposition 13 of [32], the map is dynamically coherent and plaque expansive. By [19, 21], the quasi-shadowing property and the topological quasi-stability hold for . So by Theorem 2, we obtain that, for any partially hyperbolic diffeomorphism with a uniformly compact center foliation, every nonatomic Borel measure is topologically quasi-stable.
For any diffeomorphism with a uniformly compact foliation , we can also introduce a similar definition for Definition 3.

Definition 9. Let be a diffeomorphism on with a uniformly compact foliation , we say that has the quasi-shadowing property if for any , there exists such that for any -pseudo orbit with , there exists an -W-foliation pseudo-orbit of -shadowing in the following sense:Moreover, reduce if necessary we can getwhere is the Hausdorff distance given byfor closed subsets . Note thatso in this case we also call the leaf of point -shadows the pseudo orbit .
We continue to study topologically quasi-stable measures with respect to the foliation of . We have known that if is topologically quasi-stable, then for every , there is such that for every homeomorphism that is -close to one has that every -orbit can be -shadowed by an -W-foliation pseudo-orbit of . For every topologically quasi-stable measure, we will discuss and give the measure version of above property.

Proposition 1. For every topologically quasi-stable measure with respect to the foliation of and there is such that for every homeomorphism that is -close to one has that the -orbit of can be -shadowed by an --foliation pseudo-orbit of for -a.e. . Particularly, if has a uniformly compact foliation , the -orbit of can be -shadowed by the leaf of some point for -a.e. .

Proof. For any , we assume that be given by the topological quasi-stability of . Choose a homeomorphism satisfying . For the , there exist and a family of -motions witnessing the definition of topological quasi-stability for . It can be seen that there exists a sequence with such that for -a.e. . Then, and so for all , completing the proof.
By the definition of measure expansivity, each nonatomic measure is expansive with respect to any expansive homeomorphism. We note that, for some plaque expansive diffeomorphism with respect to a continuous foliation , there may exist some nonatomic measure is not plaque expansive. In order to establish the relationship between plaque expansive diffeomorphism and plaque expansive measure, we will introduce the following definition of the local-leaf support.
The local-leaf support of a Borel measure for the foliation of is the set of all points such that for any . Hence, we can say that is a local-leaf supporting point of if . When , since is compact, we can find an such that for any .
It is easy to prove that if is plague expansive with respect its foliation , then every measure satisfying is plague expansive with respect the foliation by definition. Note that if the foliation is compact and for the foliation of , then there is at least one leaf such that . And if the foliation is compact and for , then for any .
In fact, if the foliation of has the uncountable leaves on , then the measures without local-leaf support exist and are even prevalent in a certain sense. From the topological point of view, the residual sets are the large sets, and sets of first category are the small ones. Next, we will prove that the set of measures without local-leaf support is a residual set (i.e., intersection of countably many open and dense subsets) in .

Proposition 2. If the leaves of foliation for is uncountable and has no isolated leaves, then the set of measures without local-leaf support is a residual set .

For any set , denote . For any set A which consists of whole leaves, i.e., , we call it a leaf set or a leaf-saturated set ([22]). We say that a leaf is an isolated leaf if there exists an open leaf set satisfies such that . We shall prove the above proposition as in the same manner used in Proposition 2.16 of [33].

Proof. Assume that has no isolated leaves. For , we denote thatLet us prove that is closed. Indeed, let be a sequence in converging to some . Note that each corresponds an with , and let be a limit point of the . By Proposition 2.7 in [33], for all closed , . So, for every closed neighborhood of , one has . It follows that , and hence, . Next, we prove that is nowhere dense. Indeed, we can find that every neighborhood contains these measures which can be obtained by “splitting” every -local-leaf supporting point of measure into several local-leaf supporting points lying very nearby and having each local-leaf measure . Therefore, the set of measures having local-leaf supporting points is of first category in the compact metric space . This proves the result.

3.2. Topologically Quasi-Stable Measures and Minimal Homeomorphisms

In this subsection, we will give the following results about topologically quasi-stable measures and minimal homeomorphisms. We say that a point is a transitive point if -orbit of is dense in . We say that is minimal if each -orbit is dense in . For a given Borel measure , we also say that is -transitive if the set of transitive points has positive measure with respect to .

Let have a continuous foliation . Recall that a leaf is a periodic leaf with period if . We say that a leaf is leaf nonwandering ([22]) if for any open leaf set which contains , there is such that . We denote the leaf nonwandering set of by

It is easy to see that is a closed invariant leaf set. Also we denote by the nonwandering set of . Clearly, .

Lemma 2. Let be a diffeomorphism of a compact Riemannian manifold with a continuous foliation . If there is a topologically quasi-stable measure with respect to such that can be approximated by -transitive homeomorphisms, then .

Proof. Give any and . For this and the given topologically quasi-stable measure , by Proposition 1, there exists . By the condition in this lemma, there exists a -transitive homeomorphism satisfying . Note that Proposition 1 provides a Borelian B with such that the -orbit of any can be -shadowed by an --foliation pseudo-orbit of . Because is -transitive, there exists such that . Since , there is a sequence with such thatBecause , there is a sequence such that as . As is compact, we can assume that for some . By continuity of the foliation , we get . SinceLetting , we have . As , we get . As and are arbitrary, we have . Since is closed, we prove .
In fact, every minimal diffeomorphism is -transitive for every Borel measure . By Lemma 2, we obtain the following result.

Theorem 3. Every diffeomorphism with a continuous foliation and a topologically quasi-stable measure which can be approximated by minimal homeomorphisms satisfies .

Another result relating topologically quasi-stable measures and minimal homeomorphisms is given by the following result. For a given Borel measure , we say that has -periodic points if .

Theorem 4. Let be a minimal diffeomorphism and be plaque expansive with respect to the compact foliation . If can be approximated by a homeomorphism with -periodic points, then has no topologically quasi-stable measure satisfying .

Proof. Let be a Borel measure on satisfying . Suppose by contradiction that can be approximated by a homeomorphism with -periodic points and has the topologically quasi-stable measure .
Let be the plaque expansiveness constant of . Fix and let be the constant given by the topological quasi-stability of .
By the condition in this theorem, there is a homeomorphism with -periodic points and . Because , there exist an upper semicontinuous compact-valued map having measurable domain of and a family of -motions which move sets along the foliation satisfying , and . The homeomorphism has -periodic points, which implies . As and , so does not belong to , and hence, there exists . Let be the period of as a periodic point of . By the definition of topological quasi-stability for , there exists a sequence with such that . Since is plaque expansive with respect to the foliation , we have . But also because is a periodic point of with period , we also have and , where . DefineClearly, is compact and -invariant, that is, . Because , we can get . Since , we can take . As is compact and -invariant, we obtain that is contained in . Since is minimal, the is dense in . Through the above statement, we get and . This is a contradiction which shows the result.
Under the assumptions of Theorem 4, does have a topologically quasi-stable measure satisfying ? As a partial answer to this question, we get from the following theorem that the irrational circle rotations which are minimal and plaque expansive diffeomorphisms with respect to its compact foliation both have topologically quasi-stable measures with the nonempty local-leaf support and can be approximated by periodic homeomorphisms.

Theorem 5. The circle rotations have topologically quasi-stable measures.

Proof. Clearly, the circle rotations have only one leaf which is and only have the measure with . We want to prove that any nonatomic Borel measure satisfying is a topologically quasi-stable measure for the circle rotations.
Let be the rotation of angle of the circle. When is irrational, clearly, is plaque expansive with respect to the compact foliation . Let be a nonatomic Borel measure satisfying for . For any , choose . Fix a homeomorphism satisfying , and we can define the set-valued map of as follows: . Clearly, it is a compact-valued map of . Then, and so measurable. Since for every , . As , so is upper semicontinuous. What’s more, as is nonatomic, for each showing . It follows from the definition of we also have . Finally, we prove that there is an -motion along the foliation such that . Indeed, because , there exists an -motion along the foliation from to and denoted by . Then, is -invariant and in . This proves the result.
Through a similar discussion as above, the conclusion also holds for rational rotations. This completes the proof.

3.3. Invariance on Topologically Quasi-Stable Measures

We call that a pair of diffeomorphism are topologically conjugated with respect to their continuous foliations and if there is a homeomorphism such that and for . We say that and have the same topologically quasi-stable measures if there is a bijective correspondence between the set of topologically quasi-stable measures of and the corresponding set for .

Let be a continuous map and be a Borel measure of , the pullback measure can be defined by . The following theorem is a discussion of this pullback measure.

Theorem 6. Let a pair of diffeomorphism be topologically conjugated with respect to their uniformly compact foliations and . Then, they have the same topologically quasi-stable measures.

Proof. Since are topologically conjugated with respect to their uniformly compact foliations and , there is a homeomorphism such that . In order to prove that they have the same topologically quasi-stable measures, we just have to prove that if is a topologically quasi-stable measure of , so is a topologically quasi-stable measure of .
Fix . Since , and are compact, we have that is uniformly continuous so there is such thatFor this , let be given by the topological quasi-stability with respect to . Since is compact, is uniformly continuous, and there is such thatTake a homeomorphism that is -close to . By (25), we get a homeomorphism that -close to . Since is a topologically quasi-stable measure of , for this , there is an upper semicontinuous compact-valued map of with measurable domain satisfying and a family of -motions which move sets along the foliation such thatDefine . So is an upper semicontinuous compact-valued map of . Since , we have . So has measurable domain. Moreover, we can getNote that for each , so we have . In addition, since , we have and soFinally, by (26), we can get . So we haveBy (24), we haveThen, there exists a family of -motions which move sets along the foliation such that . Note that and , so we haveThis completes the proof.

Data Availability

The data supporting the current study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

L. Wang and X. Wang are supported by the NSFC (no. 11771118, 11801336). The first author is also supported by the Applied Basic Research Program of Shanxi Province (No. 201901D211417), and the second author is also supported by the STU Scientific Research Initiation Grant (SRIG, no. NTF22020).

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