Filtrated Pseudo-Orbit Shadowing Property and Approximately Shadowable Measures

: In this paper, it is proved that every diffeomorphism possessing the ﬁltrated pseudo-orbit shadowing property admits an approximately shadowable Lebesgue measure. Furthermore, the C 1 -interior of the set of diffeomorphisms possessing the ﬁltrated pseudo-orbit shadowing property is characterized as the set of diffeomorphisms satisfying both Axiom A and the no-cycle condition. As a corollary, it is proved that there exists a C 1 -open set of diffeomorphisms, any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure.


Introduction
The notion of pseudo-orbits appears often in the several branches of the modern theory of dynamical systems; especially, the pseudo-orbit shadowing property usually plays an important role in the investigation of stability theory and ergodic theory. Let (X, d) be a compact metric space, and let f : X → X be a homeomorphism. For δ > 0, a sequence of points Denote by f |A the restriction of f to a set A ⊂ X. Let Λ ⊂ X be a closed set (not necessarily f -invariant). We say that f |Λ has the shadowing property if for every > 0 there is δ > 0 such that for any n ∈ N and δ-pseudo-orbit {x i } n−1 i=0 ⊂ Λ of f there is y ∈ X -shadowing the pseudo-orbit-that is, d( f i (y), x i ) < for all 0 ≤ i ≤ n − 1. Note that only δ-pseudo-orbits of f "contained in Λ" can be -shadowed, but the shadowing point y ∈ X is "not necessarily" contained in Λ. We say that f has the shadowing property if X = Λ in the above definition. Since X is compact, it is not difficult to show that if f |Λ has the shadowing property, then every pseudo-orbit {x i } ∞ i=−∞ ⊂ Λ can be shadowed by some true orbit of f .
In [1], we introduced the notion of shadowable measures as a generalization of the shadowing property from the measure theoretical view point, and investigated the dynamics of diffeomorphisms satisfying the notion (in fact, the dynamics of the C 1 -interior of the set of diffeomorphisms possessing the shadowable measures is characterized as uniform hyperbolicity-see [1], Theorems 1 and 2). Every dynamical system possessing the shadowing property admits shadowable measures, but the converse is not generally true. In fact, an example of a diffeomorphism g is constructed on the 2-torus T 2 such that g does not have the shadowing property but admits a shadowable Lebesgue measure (see [1], Example 3).
In this paper, generalizing the dynamics and shadowable Lebesgue measure of this example on T 2 , we introduce the notion of the filtrated pseudo-orbit shadowing property and that of approximately shadowable measures, and we prove that every diffeomorphism having the property admits an approximately shadowable Lebesgue measure. Furthermore, the C 1 -interior of the set of diffeomorphisms possessing the filtrated pseudo-orbit shadowing property is characterized as the set of diffeomorphisms satisfying both Axiom A and the no-cycle condition. Finally, by making use of a quasi-Anosov diffeomorphism, we construct a C 1 -open set of diffeomorphisms, any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure.

Definitions and Statement of the Results
Recall that (X, d) is a compact metric space and f : X → X is a homeomorphism of X. For given points x, y ∈ X and δ > 0, we write x δ y if there is a δ-pseudo-orbit {x i } n i=0 of f such that x 0 = x and x n = y for some n = n δ ∈ N. Write x δ y if x δ y and y δ x. Finally, we write x y if x δ y for any δ > 0. The chain recurrent set of f , denoted by R( f ), is the set of points x ∈ X such that x x. The chain recurrent set is one of the main subjects to consider in the shadowing theory of dynamical systems. Clearly, where Ω( f ) is the non-wandering set of f . Let X n = X × · · · × X (the n-times of direct product) be the sequences of points of X with length n ∈ N, and denote by M(X) the space of Borel probability measures of X. For any µ ∈ M(X) (not necessarily f -invariant), let µ n = µ × · · · × µ (n-times) be the direct product measure of X n . For any δ > 0, denote by P O(δ, n) the space of δ-pseudoorbits {x i } n−1 i=0 ∈ X n of f , and for > 0, denote by SP O(δ, , n) (⊂ P O(δ, n)) the set of δ-pseudo-orbits -shadowed by some point.
We say that µ ∈ M(X) is a shadowable measure of f (or simply, f is µ-shadowable) if for any > 0 there exists δ > 0 such that for any n ∈ N (if A is a subset of X, then we define the shadowable measure for f |A by the same manner). Observe that if f has the shadowing property, then f is µ-shadowable for any µ ∈ M(X). Denote by supp(µ) the support of µ ∈ M(X). Then, since X is compact, it is known that if f is µ-shadowable, then f |supp(µ) has the shadowing property (see [1], Lemma 1).
In this paper, we generalize the notion of shadowable measures to describe the dynamics of the system such as ( [1], Example 3) from the measure theoretical view point. Let µ ∈ M(X), and let Y ⊂ X be a Borel set. For any Borel set A ⊂ X, we put Hereafter, let M be a closed C ∞ manifold, and let d be a distance on M induced from a Riemannian metric · on the tangent bundle TM. Denote by Diff(M) the space of diffeomorphisms of M endowed with the C 1 -topology as usual. We say that a sequence Here intA denotes the interior of a set A ⊂ M.
We say that f has the filtrated pseudo-orbit shadowing property if there exists a filtration Denote by F S the set of f ∈ Diff(M) having the filtrated pseudo-orbit shadowing property. The first result of this paper is the following.
all the properties (a), (b) with respect to g and has the filtrated pseudo-orbit shadowing property. More precisely, for any g ∈ U ( f ), and for all x ∈ Λ and n ≥ 0. Suppose that Λ is hyperbolic. Then it is well-known that f |Λ has the shadowing property (see [2,3]).
The stable manifold of a point x ∈ Λ is the set The unstable manifold, W u (x), of x ∈ Λ is also defined analogously for n → −∞. It is also well-known that W s (x) and W u (x) are both immersed manifolds (see [3], among others).
We say that f is Anosov when the whole space M is hyperbolic. At this moment, let us remark that any µ ∈ M(M) is shadowable if f is Anosov, and thus, every Anosov diffeomorphism admits a shadowable Lebesgue measure m such that supp(m) = M.
Hereafter, let P( f ) be the set of periodic points of f , and recall that Ω( f ) is the set of non-wandering points of it. We say that f satisfy Axiom A if Ω( f ) is hyperbolic and Ω( f ) = P( f ). Let f satisfies Axiom A. Then the non-wandering set has the so-called spectral decomposition-that is, We say that f has a cycle if there is a subsequence {Λ i j ( f )} l j=1 (2 ≤ l ≤ L + 1) of the spectral decomposition such that Note that if f satisfies the no-cycle condition, then R( f ) = Ω( f ) (see [4]).
The next result is the following.

Theorem 2. Let f ∈ Diff(M). Then f ∈ intF S if and only if f satisfies both Axiom A and the no-cycle condition.
We say that f is quasi-Anosov if for any v ∈ TM \ {0}, { D f n (v) : n ∈ Z} is unbounded. In [5], quasi-Anosov diffeomorphisms are characterized as diffeomorphisms satisfying both Axiom A and the no-cycle condition such that for any x ∈ M, Every Anosov diffeomorphism is quasi-Anosov, but an example of quasi-Anosov, non-Anosov diffeomorphism is constructed by [6] (for more information, see [7]). In this paper, by making use of a quasi-Anosov diffeomorphism, we construct a C 1 -open set of Diff(M), any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure (see Corollary 1).
For quasi-Anosov diffeomorphisms, the relationship to the shadowing property is considered in [8], and the following result is obtained therein.

Theorem 3. Let f ∈ Diff(M). Then f is quasi-Anosov possessing the shadowing property if and only if f is Anosov.
Since every quasi-Anosov diffeomorphism is in intF S by Theorem 2, the next result follows from Theorems 1 and 3.

Remark 2.
Since the example g on T 2 constructed in ( [1], Example 3) is in intF S, every h C 1 -nearby g admits an approximately shadowable Lebesgue measure. However, it is easy to see that, for any C 1 -neighborhood V (g) of g, there is a h ∈ V (g) possessing the shadowing property.
We close this section by pointing out an example which does not admit an approximately shadowable Lebesgue measure. Example 1. Let S 1 = {e 2πiθ : θ ∈ R} ⊂ C. For α ∈ R \ Q, let ρ α : S 1 → S 1 be an irrational rotation map defined by ρ α (e 2πiθ ) = e 2πi(θ+α) . Then the map ρ α does not admit an approximately shadowable Lebesgue measure since ρ α does not satisfy the shadowing property (see [1], Example 2) and we have Y = S 1 for ∅ = Y ⊂ S 1 with f (Y) ⊂ Y.

Proofs of the Results
In this section, we give the proofs of Theorems 1 and 2 and Corollary 1.
Proof of Theorem 1. Suppose that f ∈ F S, and let m ∈ M(M) be the normalized Lebesgue measure on M. Let ∅ = M 0 ⊂ M 1 ⊂ · · · ⊂ M K = M be a filtration adapted to f as in the definition of the filtrated pseudo-orbit shadowing property, and recall the conditions (a) and (b) that f meets. We define a stable set for is an increasing sequence of closed sets satisfying that for 1 (3) and (6), for > 0 there exists N ∈ N such that In what follows, we put (4)). Now, let us take δ 1 > 0 such that if x ∈ W N k and y ∈ W N l for k = l, then d(x, y) > δ 1 (see (5) above). Thus we have the following Claim. There exists 0 < δ 2 < δ 1 such that for any δ-pseudo- Indeed, let {x i } n i=0 ⊂ Y be a given δ-pseudo-orbit of f with 0 < δ < δ 1 and n ≥ N + 1. Then it is easy to see that by the choice of by the uniform continuity of f , we can choose 0 < δ 2 < δ 1 such that if 0 < δ < δ 2 , then x i ∈ intM k for N + 1 ≤ i ≤ n. Moreover, by (4), we have x i ∈ int(M k \ M k−1 ) for N + 1 ≤ i ≤ n, and thus the claim is proved.
Finally, let us show that every pseudo-orbit {x i } n i=0 ⊂ Y is shadowed by a true orbit of f . For > 0, by the uniform continuity of f , we can choose 0 < δ < δ 2 such that every δ -pseudo-orbit of f with length less than N + 1 can be -shadowed by a true orbit of f . Thus, it is not difficult to show that any δ -pseudo-orbit {x i } n i=0 ⊂ Y of f can be shadowed by a true orbit of f reducing δ if necessary. Therefore, for the set Y, if we definẽ m ∈ M(M) asm for any Borel set A ⊂ M, thenm is shadowable, m(Y) > 1 − and f (Y) ⊂ Y, and thus Theorem 1 is proved.
We need a lemma to prove Theorem 2. Remark that if f possesses the filtrated pseudoorbit shadowing property, then f |R( f ) has the shadowing property by definition. The following lemma proved in ( [9], Proposition 2.3) will be used in the proof of the "only if" part of Theorem 2.

Lemma 1.
If f |R( f ) has the shadowing property, then the shadowing point can be taken from Ω( f ) for any pseudo-orbit in Ω( f ).

Proof of Theorem 2.
To prove the if part of the theorem, suppose that f satisfies both Axiom A and the no-cycle condition. Then [3,4]). To prove the filtrated pseudoorbit shadowing property for f , we note that for any 1 ≤ k ≤ K there is a neighborhood U k of Λ k ( f ) with the property that for all > 0 there exists δ > 0 such that for any δ-pseudo- Thus, we can see that there is δ 3 > 0 such that for any δ-pseudo- On the other hand, by the uniform continuity of f , every pseudo-orbit of f with length less than 2m k + 1 can be shadowed by a true orbit of f . Therefore, it is not difficult to show that any δ-pseudo-orbit {x i } n i=0 ⊂ int(M k \ M k−1 ) (1 ≤ k ≤ K) of f can be shadowed by a true orbit of f reducing δ if necessary.
Finally, it can be checked that since f is R-stable (and thus, is Ω-stable), any g C 1nearby f also meets all of the conditions (a), (b) with respect to ∅ = M 0 ⊂ M 1 ⊂ · · · ⊂ M K = M (see [3], pp. 435-444) and has the filtrated pseudo-orbit shadowing property, and thus, the if part is proved.
To prove the only if part, let us denote by ΩS the set of diffeomorphisms such that f : Ω( f ) → Ω( f ) has the shadowing property; and - The shadowing point can be taken from Ω( f ).
It was shown in ( [10], Proposition 1) that any f in the C 1 -interior of ΩS satisfies both Axiom A and the no-cycle condition. Thus, to get the conclusion, it is enough to show that if f ∈ F S, then f is in ΩS. Suppose that f ∈ F S. Then f |R( f ) has the shadowing property, and thus, by Lemma 1, f is in ΩS since the shadowing point can be taken from Ω( f ). Thus, Theorem 2 is proved.
Proof of Corollary 1. Let f be a quasi-Anosov diffeomorphism, so that f satisfies both Axiom A and the no-cycle condition. Since f ∈ intF S by Theorem 2, every g C 1 -nearby f admits an approximately shadowable Lebesgue measure by Theorem 1.
It is proved that every g C 1 -nearby f is also quasi-Anosov by ( [5], Lemma 1. 6), and that f is Anosov if and only if W s (p) is the same dimension for all p ∈ P( f ) by ( [5], Corollary 1).
Suppose further that f is not Anosov. Then, since there are hyperbolic periodic points p, q ∈ P( f ) with different indices, that is, dim W s (p) = dim W s (q), every g C 1 -nearby f also has periodic points with different indices. Thus, g is quasi-Anosov but not Anosov by ([5], Corollary 1), so that g does not have the shadowing property by Theorem 3.
Author Contributions: The authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.