Asymptotic measure-expansiveness for generic di ff eomorphisms

∀ ∈ i , then = x y. Generally speaking, expansiveness means that if any two real orbits are separated by a small distance, the two orbits are identical, and therefore it is appropriate for studying smooth dynamic systems. Expansivities are hence a valuable notion in the investigation of hyperbolic structures (see [2–17], etc.). Mañé [17] proved that aC1 robustly expansive diffeomorphism f is quasi Anosov, i.e., the set {∥ ( )∥ ∈ } Df v n : n


Introduction
Throughout this paper, we will assume M to be a compact smooth manifold and d to be the distance on M induced by a Riemannian metric ∥⋅∥. We also assume → f M M : to be a diffeomorphism and denote by ( ) M Diff the set of diffeomorphisms of M endowed with C 1 topology. It is to note that expansiveness has been earlier suggested in the study by Utz [1]. A diffeomorphism f is said to be expansive if there exists a positive constant > δ 0 such that for any two points i i ∀ ∈ i , then = x y. Generally speaking, expansiveness means that if any two real orbits are separated by a small distance, the two orbits are identical, and therefore it is appropriate for studying smooth dynamic systems. Expansivities are hence a valuable notion in the investigation of hyperbolic structures (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], etc.). Mañé [17] proved that a C 1 robustly expansive diffeomorphism f is quasi Anosov, i.e., the set {∥ ( )∥ ∈ } Df v n : n is unbounded for all ∈ ⧹ { } v TM 0 . Morales and Sirvent [18] introduced stochastic perspectives of expansiveness, called measure-expansiveness. Let us assume ( ) M to be the set of all Borel probability measures on M endowed with the weak * topology and ( ) * M to be the set of nonatomic measures ∈ ( ) μ M . It is known that , then we say that a diffeomorphism f is measure expansive.
Here, δ is called a measure expansive constant of f . Now, we introduce a general notion of measureexpansiveness called the asymptotic measure expansive (see [19,Example 1.1]). The notion was suggested in [19]. Let us assume that ∈ , then we say that f is asymptotic measure expansive.
The following notion is suggested in [20]. A diffeomorphism f on M is said to be continuum-wise expansive if there is a positive constant > δ 0 such that, for any nontrivial compact connected set Λ, there is an integer ∈ n such that , : , Λ for any subset ⊂ M Λ and λ is nontrivial, which means that Λ is neither one point nor one orbit.
Regarding the result of Artigue and Carrasco-Olivera [21], it is observed that a diffeomorphism f is measure-expansive if it is continuum-wise expansive. However, the converse is not true. We already know that a diffeomorphism f is measure-expansive if it is asymptotic measure-expansive. Here, the converse is also untrue. Therefore, we have a question: What is relation between asymptotic measure expansiveness and continuum-wise expansiveness?
, then a diffeomorphism f is said to be Anosov.
It is known that if Λ is hyperbolic for f , then Λ is expansive, thus it is measure-expansive and asymptotic measure-expansive. Per f Ω is hyperbolic. According to Aoki [22] and Hayashi [23], f satisfies Axiom A and has no-cycles if f is star.
In this paper, we consider sets of diffeomorphisms that are residual for the Baire category, i.e., sets that contain a countable intersection of dense and open subsets of ( ) M Diff . Regarding C 1 generic diffeomorphisms, it is known that the periodic points are dense in ( ) f Ω by Pugh's closing lemma [24]. Using the C 1 generic property, Arbieto proved in [25] that f satisfies Axiom A and has no-cycles for a C 1 generic expansive diffeomorphism. Lee [26] proved that a C 1 generic measure expansive diffeomorphism f satisfies Axiom A and has no-cycle. Recently, Lee [27] proved that f satisfies Axiom A and has no-cycles for a C 1 generic continuum-wise expansive diffeomorphism. According to the abovementioned results, we consider general concepts of measure expansiveness. The following is the primary theorem of the paper.
, f satisfies Axiom A and has no-cycles if it is asymptotic measureexpansive.
For any hyperbolic periodic point p, define the following sets . We write p q . It is clear that :~, which is called the homoclinic class. It is known that the set is a closed f -invariant and transitive set. Note that if a diffeomorphism f satisfies Axiom A, then the nonwandering set ( ) f Ω is a disjoint union of transitive invariant closed subsets. In fact, these sets are homoclinic classes that each contain a hyperbolic periodic point. Several researchers are studying these sets and their hyperbolicity (see [4,[28][29][30][31][32][33][34], etc.). We study whether the homoclinic class is hyperbolic using the asymptotic measure-expansiveness. Yang and Gan [34] proved that a homoclinic class ( ) H p f , is hyperbolic if it is expansive for a C 1 generic diffeomorphism f. Koo et al. [35] proved that a locally maximal homoclinic class ( ) , is hyperbolic if it is measure-expansive for a C 1 generic diffeomorphism f . The result is a general version of the proof in [35]. In [31], Lee proved that a homoclinic class ( ) , is hyperbolic if it is continuum-wise expansive for a C 1 generic f . According to the results, we prove the following:

Proof of Theorem A
Theorem A will be proven in this section, which requires some notions to be taken into account. A point ∈ ( ) p Per f is weak hyperbolic if there is gC 1 close to f such that the derivative map ( ) D g p π p has an eigenvalue λ with | | = λ 1. For any > ε 0, we consider a closed curve η to be ε simply periodic if η satisfies the following conditions: , and (c) η is normally hyperbolic (see [34]).
If a ∈ ( ) p Per f is hyperbolic, then there are a C 1 neighborhood ( ) f of f and a locally maximal neighborhood U of p such that there exists the hyperbolic periodic = ⋂ ( ) ∈ p g U g n n for any ∈ ( ) g f . Here, p g is called a continuation.
The following is called Franks' lemma [36], which is a useful notion for a C 1 robust property.
If a diffeomorphism f has a weak hyperbolic periodic point, then for any neighborhood ( ) f of f and any > ε 0, there are ∈ ( ) g f and a small curve with the following property: is g periodic, i.e., there is ∈ n such that ( ) = g n ; (b) the length of ( ) g i is less than ε for all ∈ i ; (c) the endpoints of the curve are hyperbolic; (d) is normally hyperbolic with respect to g (see [37]).
Proof. Let us assume p to be a weak hyperbolic periodic point of f and ( ) f to be a C 1 neighborhood of f . For simplicity, we may assume that ( ) = f p p. According to Lemma 2.1, there is ∈ ( ) g f such that ( ) = g p p and the derivative map D g p has an eigenvalue λ with | | = λ 1, i.e., g has a non hyperbolic periodic point p. As given in the proof of [26, Lemma 2.2], hC 1 can be found close to g (also, ∈ ( ) h f ) such that (i) ( ) = h k for some ∈ k , and (ii) | → h : k is the identity map.
In items (i) and (ii), = k 1 and = k 2 if the eigenvalue λ is a positive or negative real number, respectively. If the eigenvalue λ is a complex number, then one can take > l 0 such that = k l. As in the proof of [26, Lemma 2.2], it is clear that is normally hyperbolic and the length of is less than ε. Therefore, the small closed curve satisfies items (a), (b), and (d). Finally, we show item (c). Let us assume that q and r are the endpoints of the closed curve . For simplification, we assume that , we assume that f is asymptotic measure-expansive. According to Aoki [22] and Hayashi [23], it is sufficient to show that f is a star. Suppose, by contradiction, that f is not a star. If f is not a star, f has a simple periodic curve with hyperbolic endpoints according to Lemma 2.3 , the endpoints of the curve are hyperbolic, and is normally hyperbolic. Let us now assume ν to be a normalized Lebesgue measure on . ∈ ( ) μ M is defined as . It can therefore be seen that ( , it is observed that kn for all ∈ n . Therefore, we know that n n ( ( ( ))) → μ f δ x Γ , 0 n as → ∞ n because f is asymptotic measure-expansive. This is a contradiction because ( ( ( ))) → ( → ∞) μ f δ x n Φ , ̸ 0 n . Therefore, for C 1 generic f , f satisfies both Axiom A and the no-cycle condition if f is asymptotic measure-expansive. □

Proof of Theorem B
Theorem B will be proven in this section using various results of a C 1 generic property. For any > δ 0, we consider a point p to be a δ weak hyperbolic periodic point if where λ is the eigenvalue λ of D f p , and ( ) π p is the period of p. We consider f to be Kupka-Smale if every periodic point is hyperbolic and its stable and unstable manifolds are transversal intersections. It is well known that a diffeomorphism f is a residual subset of ( ) M Diff if it is Kupka-Smale (see [38]).
For any > ε 0, a sequence of points : . It is clear that Lemma 3.1. There is a residual subset 1 in ( ) M Diff such that, for given ∈ f 1 , we have the following: (a) f is Kupka-Smale (see [38]); (b) for any > δ 0 and any ∈ ( ) p Per f , there exists ∈ ( ) g f for any C 1 neighborhood ( ) f of f such that g has a δ simply periodic curve , where the two endpoints of are homoclinically related to p g . Therefore, f has a δ simply periodic curve such that the two endpoints of are homoclinically related to p (see [34]); , (see [39]).
and > δ 0 to be given. If q is a δ weak hyperbolic periodic point for f , there exists gC 1 close to f such that g has a δ simply periodic curve with endpoints that are homoclinically related to p g .
Proof. See the proof of [33,34]. Proof. For ∈ f 1 , assume that f is asymptotic measure-expansive. We shall derive a contradiction. Suppose that there is ∈ ( ) ∩ ( ) q H p f Per f , such that q is weak hyperbolic. Therefore, there is gC 1 close to f such that g has a δ simply periodic curve with endpoints that are homoclinically related to p g according to Lemma 3.2. Because ∈ f 1 , f has a δ simply periodic curve such that the two endpoints of are homoclinically related to p by Lemma 3.1.
Assume that the period of is > L 0. Let us assume that μ be a normalized Lebesgue measure on .
. Therefore, we know that We assume p to be a hyperbolic periodic point of f having a period of ( ) π p . Therefore, if … μ μ μ , , , n

Conflict of interest:
Author states no conflict of interest.