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The ergodic shadowing property from the robust and generic view point
Advances in Difference Equations volume 2014, Article number: 170 (2014)
Abstract
In this paper, we discuss that if a diffeomorphisms has the {C}^{1}stably ergodic shadowing property in a closed set, then it is a hyperbolic elementary set. Moreover, {C}^{1}generically: if a diffeomorphism has the ergodic shadowing property in a locally maximal closed set, then it is a hyperbolic basic set.
MSC:34D30, 37C20.
1 Introduction
Let M be a closed {C}^{\mathrm{\infty}} manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the {C}^{1}topology. Denote by d the distance on M induced from a Riemannian metric \parallel \cdot \parallel on the tangent bundle TM. Let f\in Diff(M). For \delta >0, a sequence of points {\{{x}_{i}\}}_{i=a}^{b} (\mathrm{\infty}\le a<b\le \mathrm{\infty}) in M is called a δpseudo orbit of f if d(f({x}_{i}),{x}_{i+1})<\delta for all a\le i\le b1. For given x,y\in M, we write x\u21ddy if for any \delta >0, there is a δpseudo orbit {\{{x}_{i}\}}_{i=a}^{b} (a<b) of f such that {x}_{a}=x and {x}_{b}=y. Let Λ be a closed finvariant set. We say that f has the shadowing property in Λ if for every \u03f5>0 there is \delta >0 such that, for any δpseudo orbit {\{{x}_{i}\}}_{i=a}^{b}\subset \mathrm{\Lambda} of f (\mathrm{\infty}\le a<b\le \mathrm{\infty}), there is a point y\in \mathrm{\Lambda} such that d({f}^{i}(y),{x}_{i})<\u03f5 for all a\le i\le b1. If \mathrm{\Lambda}=M, then f has the shadowing property. The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory. For instance, Sakai [1] proved that if f has the {C}^{1}robustly shadowing property, then f is structurally stable. Now we introduce the notion of the ergodic shadowing property which was introduced and studied by [2]. Lee has shown in [3] that if f belongs to the {C}^{1}interior of the set of all diffeomorphisms having the ergodic shadowing property, then it is structurally stable diffeomorphisms. In [4], Lee showed that if f is local star condition and has the ergodic shadowing property on the homoclinic class, then it is hyperbolic. For any \delta >0, a sequence \xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}} is a δergodic pseudo orbit of f if for N{p}_{n}^{+}(\xi ,f,\delta )=\{i:d(f({x}_{i}),{x}_{i+1})\ge \delta \}\cap \{0,1,\dots ,n1\}, and N{p}_{n}^{}(\xi ,f,\delta )=\{i:d({f}^{1}({x}_{i}),{x}_{i1})\ge \delta \}\cap \{n+1,\dots ,1,0\}
Here #A is the number of elements of the set A. We say that f has the ergodic shadowing property in Λ (or f{}_{\mathrm{\Lambda}} has ergodic shadowing) if for any \u03f5>0, there is a \delta >0 such that every δergodic pseudo orbit \xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda} of f there is a point z\in \mathrm{\Lambda} such that, for N{s}_{n}^{+}(\xi ,f,z,\u03f5)=\{i:d({f}^{i}(z),{x}_{i})\ge \u03f5\}\cap \{0,1,\dots ,n1\}, and N{s}_{n}^{}(\xi ,f,z,\u03f5)=\{i:d({f}^{i}(z),{x}_{i})\ge \u03f5\}\cap \{n+1,\dots ,1,0\},
Note that f has the ergodic shadowing property on Λ and f has the ergodic shadowing property in Λ are different notions. That is, the shadowing point is in M or Λ. In the first notion, the shadowing point is in M. In the second notion, the shadowing point is in Λ. In this paper we consider the latter case.
We say that Λ is locally maximal if there is a compact neighborhood U of Λ such that
Now, we introduce the notion of the {C}^{1}stably ergodic shadowing property in a closed set.
Definition 1.1 Let Λ be a closed finvariant set. We say that f has the {C}^{1}stably ergodic shadowing property in Λ if

(i)
there is a neighborhood U of Λ and a {C}^{1}neighborhood \mathcal{U}(f) of f such that {\mathrm{\Lambda}}_{f}(U)=\mathrm{\Lambda}={\bigcap}_{n\in \mathbb{Z}}{f}^{n}(U) (that is, Λ is locally maximal);

(ii)
for any g\in \mathcal{U}(f), g has the ergodic shadowing property on {\mathrm{\Lambda}}_{g}(U)={\bigcap}_{n\in \mathbb{Z}}{g}^{n}(U), where {\mathrm{\Lambda}}_{g}(U) is the continuation of Λ.
We say that Λ is hyperbolic if the tangent bundle {T}_{\mathrm{\Lambda}}M has a Dfinvariant splitting {E}^{s}\oplus {E}^{u} and there exist constants C>0 and 0<\lambda <1 such that
for all x\in \mathrm{\Lambda} and n\ge 0. If \mathrm{\Lambda}=M, then f is Anosov. We say that Λ is a basic set (resp. elementary set) if f{}_{\mathrm{\Lambda}} is transitive (resp. mixing) and locally maximal. Note that if Λ is hyperbolic, then we can easily show that there is a periodic point such that the orbit of the periodic point is dense in the set. Then we get the following.
Theorem 1.2 [[5], Theorem 3.3]
Let Λ be a closed finvariant set. If f has the {C}^{1}stably ergodic shadowing property in Λ, then it is a hyperbolic elementary set.
Corollary 1.3 If f belongs to the {C}^{1}interior of the set of all diffeomorphisms having the ergodic shadowing property, then it is transitive Anosov.
We say that a subset \mathcal{G}\subset Diff(M) is residual if contains the intersection of a countable family of open and dense subsets of Diff(M); in this case is dense in Diff(M). A property P is said to be {C}^{1}generic if P holds for all diffeomorphisms which belong to some residual subset of Diff(M). We use the terminology ‘for {C}^{1}generic f’ to express ‘there is a residual subset \mathcal{G}\subset Diff(M) such that, for any f\in \mathcal{G}\dots .’ In [6], Abdenur and Díaz proved that if tame diffeomorphisms has the shadowing property, then it is hyperbolic. Still open is the question if {C}^{1}generically: f is shadowable, then is it hyperbolic?
Recently, Ahn et al. [7] have given a partial answer which is {C}^{1}generically: if a locally maximal homoclinic class is shadowing, then it is hyperbolic. Lee has shown in [8] that {C}^{1}generically: if f has the limit shadowing property on the homoclinic class, then it is hyperbolic. Inspired by this, we consider that {C}^{1}generically: f has the ergodic shadowing property in a locally maximal closed set. Then we have the following.
Theorem 1.4 For {C}^{1}generic f, if f has the ergodic shadowing property in a locally maximal closed set Λ, then it is a hyperbolic elementary set. Moreover, {C}^{1}generically: if f has the ergodic shadowing property, then it is transitive Anosov.
2 Proof of Theorem 1.4
Let P(f) be the set of periodic points of f. If f{}_{\mathrm{\Lambda}} is transitive, then every p\in \mathrm{\Lambda}\cap P(f) is saddle, that is, there is no eigenvalues of {D}_{p}{f}^{\pi (p)} with modulus equal to 1, at least one of them is greater than 1, at least one of them is smaller than 1, where \pi (p) is the minimum period of p.
Lemma 2.1 [[2], Corollary 3.5]
If f has the ergodic shadowing property in Λ, then f{}_{\mathrm{\Lambda}} is mixing.
By Lemma 2.1, f has the ergodic shadowing property in Λ, then f{}_{\mathrm{\Lambda}} is mixing, and so f{}_{\mathrm{\Lambda}} is transitive. Thus p\in \mathrm{\Lambda}\cap P(f) is neither a sink nor a source.
Lemma 2.2 [[2], Lemma 3.2]
If f has the ergodic shadowing property in Λ, then f has a finite shadowing property in Λ.
We say that f has the finite shadowing property on Λ if for any \u03f5>0 there is \delta >0 such that, for any finite δpseudo orbit \{{x}_{0},{x}_{1},\dots ,{x}_{n}\}\subset \mathrm{\Lambda}, there is y\in M such that d({f}^{i}(y),{x}_{i})<\u03f5 for all 0\le i<n. In [[9], Lemma 1.1.1], Pilyugin showed that f has a finite shadowing shadowing property on Λ, then f has the shadowing property on Λ.
Lemma 2.3 Let f have the ergodic shadowing in Λ and Λ be locally maximal in U. Then the shadowing point taken from Λ.
Proof Let f have the ergodic shadowing property in Λ, and let U be a locally maximal of Λ. For any \u03f5>0, let \delta >0 be the number of the ergodic shadowing property of f. Take a sequence \gamma ={\{{x}_{i}\}}_{i=0}^{n} (n\ge 1) such that γ is a δpseudo orbit of f and \gamma \subset \mathrm{\Lambda}. As in the proof of [[2], Lemma 3.1], there is a δpseudo orbit \eta ={\{{x}_{i}\}}_{i=n}^{0} such that \eta \subset \mathrm{\Lambda}. Then we set \xi =\{\dots ,\gamma ,\eta ,\gamma ,\eta ,\dots \} is a δergodic pseudo orbit of f. Clear that \xi \subset \mathrm{\Lambda}. Since f has the ergodic shadowing property in Λ, ξ can be ergodic shadowed by some point y\in \mathrm{\Lambda}. By Lemma 2.2, there is \gamma \in \xi such that d({f}^{i}(y),{x}_{i})<\u03f5 for 0\le i\le n1. By [[9], Lemma 1.1.1], f has the shadowing property on Λ. Since Λ is locally maximal in U, the shadowing point y\in \mathrm{\Lambda}. □
Let p\in P(f) be a hyperbolic saddle with period \pi (p)>0. Then there are the local stable manifold {W}_{\u03f5}^{s}(p) and the local unstable manifold {W}_{\u03f5}^{u}(p) of p for some \u03f5=\u03f5(p)>0. It is easily seen that if d({f}^{n}(x),{f}^{n}(p))\le \u03f5 for all n\ge 0, then x\in {W}_{\u03f5}^{s}(p), and if d({f}^{n}(x),{f}^{n}(p))\le \u03f5 for all n\le 0, then x\in {W}_{\u03f5}^{u}(p). The stable manifold {W}^{s}(p) and the unstable manifold {W}^{u}(p) defined as following. It is well known that if p is a hyperbolic periodic point of f with period k, then the sets
are {C}^{1}injectively immersed submanifolds of M.
Lemma 2.4 Let p,q\in P(f) be hyperbolic saddles. If f has the ergodic shadowing property in a closed set Λ, then {W}^{s}(p)\cap {W}^{u}(q)\ne \mathrm{\varnothing}, and {W}^{u}(p)\cap {W}^{s}(q)\ne \mathrm{\varnothing}.
Proof Let p,q\in P(f) be hyperbolic saddles, and let U be a locally maximal neighborhood of Λ. Suppose that f has the ergodic shadowing property in a locally maximal Λ. Since p and q are hyperbolic, there are \u03f5(p)>0 and \u03f5(q)>0 as in the above. Take \u03f5=min\{\u03f5(p),\u03f5(q)\}/4 and let 0<\delta \le \u03f5 be the number of the ergodic shadowing property of f. For simplicity, we may assume that f(p)=p and f(q)=q. Since f has the ergodic shadowing property in Λ, f{}_{\mathrm{\Lambda}} is chain transitive. Then we can construct a finite δpseudo orbit form p to q as follows: {x}_{0}=p, {x}_{n}=q (n\ge 1), and d(f({x}_{i}),{x}_{i+1})<\delta for all 0<i<n1. Put (i) {x}_{i}={f}^{i}(p), for all i\le 0, and (ii) {x}_{n+i}={f}^{i}(q) for all i\ge 0. Then we have the sequence \xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}=\{\dots ,p,{x}_{1},{x}_{2},\dots ,{x}_{n},{x}_{n+1},\dots \}. It is clearly a δergodic pseudo orbit of f. Since f has the ergodic shadowing property in Λ and locally maximal, by Lemma 2.2, f has the finite shadowing property on Λ and so, by [[9], Lemma 1.1.1], f has the shadowing property in Λ. By the shadowing property in Λ, we can show that Orb(y)\subset {W}^{u}(p)\cap {W}^{s}(q), and so {W}^{u}(p)\cap {W}^{s}(q)\ne \mathrm{\varnothing}. The other case is similar. □
A diffeomorphism f is KupkaSmale if their periodic points of f are hyperbolic and if p,q\in P(f), then {W}^{s}(p) is transversal to {W}^{u}(q). Then it is {C}^{1}residual in Diff(M). Denote by \mathcal{KS}(M) the set of all KupkaSmale diffeomorphisms. The following was proved by [10].
Lemma 2.5 [[10], Lemma 2.4]
Let Λ be locally maximal in U, and let \mathcal{U}(f) be given. If for any g\in \mathcal{U}(f), p\in {\mathrm{\Lambda}}_{g}(U)\cap P(g) is not hyperbolic, then there is {g}_{1}\in \mathcal{U}(f) such that {g}_{1} has two hyperbolic periodic points p,q\in {\mathrm{\Lambda}}_{{g}_{1}}(U) with different indices.
Denote by \mathcal{F}(M) the set of f\in Diff(M) such that there is a {C}^{1} neighborhood \mathcal{U}(f) of f such that, for any g\in \mathcal{U}(f), every p\in P(g) is hyperbolic. In [11], Hayashi proved that f\in \mathcal{F}(M) if and only if f satisfies both Axiom A and the nocycle condition. We say that f is the local star condition diffeomorphism if there exist a {C}^{1}neighborhood \mathcal{U}(f) and a neighborhood U of Λ such that, for any g\in \mathcal{U}(f), every p\in {\mathrm{\Lambda}}_{g}(U)\cap P(g) is hyperbolic (see [12]). Denote by \mathcal{F}(\mathrm{\Lambda}) the set of all local star diffeomorphisms. Note that there are a {C}^{1}neighborhood \mathcal{U}(f) and a neighborhood U of p such that, for all g\in \mathcal{U}(f), there is a unique hyperbolic periodic point {p}_{g}\in U of g with the same period as p and index({p}_{g})=index(p). Here index(p)=dim{E}_{p}^{s}, and the point {p}_{g} is called the continuation of p.
Lemma 2.6 [[13], Lemma 2.2]
There is a residual set {\mathcal{G}}_{1}\subset Diff(M) such that, for any f\in {\mathcal{G}}_{1}, if for any {C}^{1}neighborhood \mathcal{U}(f) of f, there exists g\in \mathcal{U}(f) such that two hyperbolic periodic points {p}_{g},{q}_{g}\in P(g) with index({p}_{g})\ne index({q}_{g}), then f has two hyperbolic periodic points p,q\in P(f) with index(p)\ne index(q).
Lemma 2.7 There is a residual set {\mathcal{G}}_{2}\subset Diff(M) such that, for any f\in {\mathcal{G}}_{2}, if f has the ergodic shadowing property in a locally maximal Λ, then for any p,q\in \mathrm{\Lambda}\cap P(f)
Proof Let f\in {\mathcal{G}}_{2}={\mathcal{G}}_{1}\cap \mathcal{KS}(M), and let p,q\in \mathrm{\Lambda}\cap P(f) be hyperbolic saddles. Suppose that f has the ergodic shadowing property in a locally maximal Λ. Then by Lemma 2.4 {W}^{s}(p)\cap {W}^{u}(q)\ne \mathrm{\varnothing} and {W}^{u}(p)\cap {W}^{s}(q)\ne \mathrm{\varnothing}. Since f\in {\mathcal{G}}_{2}, {W}^{s}(p)\u22d4{W}^{u}(q)\ne \mathrm{\varnothing} and {W}^{u}(p)\u22d4{W}^{s}(q)\ne \mathrm{\varnothing}. This means that p\sim q and so index(p)=index(q). □
Let p be a periodic point of f. For 0<\delta <1, we say that p has a δweak eigenvalue if D{f}^{\pi (p)}(p) has an eigenvalue λ such that {(1\delta )}^{\pi (p)}<\lambda <{(1+\delta )}^{\pi (p)}. We say that a periodic point has a real spectrum if all of its eigenvalues are real and simple spectrum if all its eigenvalues have multiplicity one. Denote by {P}_{h}(f) the set of all hyperbolic periodic points of f.
Lemma 2.8 [[14], Lemma 5.1]
There is a residual set {\mathcal{G}}_{3}\subset Diff(M) such that, for any f\in {\mathcal{G}}_{3}:

For any\delta >0, if for any{C}^{1}neighborhood\mathcal{U}(f)of f there existg\in \mathcal{U}(f)and{p}_{g}\in {P}_{h}(g)with a δweak eigenvalue, then there isp\in {P}_{h}(f)with a 2δweak eigenvalue.

For any\delta >0, ifq\in {P}_{h}(f)with a δweak eigenvalue and a real spectrum, then there isp\in {P}_{h}(f)with a δweak eigenvalue with a simple real spectrum.
Lemma 2.9 There is a residual set {\mathcal{G}}_{4}\subset Diff(M) such that, for any f\in {\mathcal{G}}_{4}, if f has the ergodic shadowing property in a locally maximal Λ, then there exists \eta >0 such that, for any q\in \mathrm{\Lambda}\cap {P}_{h}(f), q has no ηweak eigenvalues.
Proof Let f\in {\mathcal{G}}_{4}={\mathcal{G}}_{2}\cap {\mathcal{G}}_{3} have the ergodic shadowing property in a locally maximal Λ. We will derive a contradiction. Suppose that, for any \eta >0, there is q\in \mathrm{\Lambda}\cap {P}_{h}(f) such that q has an ηweak eigenvalue. By Franks’ lemma, there is g {C}^{1}close to f such that p is not hyperbolic. By Franks’ lemma and Lemma 2.5, there is h {C}^{1}nearby g and {C}^{1}close to f such that h has tow hyperbolic periodic points {q}_{h}, {\gamma}_{h} with different indices. Since f\in {\mathcal{G}}_{1}, and it is locally maximal, by Lemma 2.6 f has two hyperbolic periodic points q, γ in Λ. Since f has the ergodic shadowing property in Λ, this is a contradiction by Lemma 2.7. □
Proposition 2.10 There is a residual set {\mathcal{G}}_{4}\subset Diff(M) such that, for any f\in {\mathcal{G}}_{4}, if f has the ergodic shadowing property in Λ, then f\in \mathcal{F}(\mathrm{\Lambda}).
Proof Let f\in {\mathcal{G}}_{4} have the ergodic shadowing property in a locally maximal Λ. Suppose by contradiction that f\notin \mathcal{F}(\mathrm{\Lambda}). Then there are g {C}^{1}close to f and q\in P(g) such that q has a ηweak eigenvalue. Then by Lemma 2.9, we get a contradiction. Thus f\in \mathcal{F}(\mathrm{\Lambda}). □
Proposition 2.11 [[15], Proposition A]
Let f\in {\mathcal{G}}_{4}, and let Λ be locally maximal. If f has the ergodic shadowing property in Λ, then there are m>0, C\ge 1 and \lambda \in (0,1) such that, for any p\in \mathrm{\Lambda}\cap P(f) with \pi (q)>m, we have
where \pi (p) is the period of p.
Remark 2.12 By Pugh’s closing lemma, there is a residual set {\mathcal{G}}_{6}\subset Diff(M) such that, for any f\in {\mathcal{G}}_{6}, if f{}_{\mathrm{\Lambda}} is transitive, then there is a periodic orbit {p}_{n} such that Orb({p}_{n})\to \mathrm{\Lambda} in Hausdorff metric.
Lemma 2.13 [[16], Theorem 3.8]
There is residual set {\mathcal{G}}_{5}\subset Diff(M) such that, for any f\in {\mathcal{G}}_{5}, for any ergodic measure μ of f, there is a sequence of the periodic point {p}_{n} such that {\mu}_{{p}_{n}}\to \mu in weak^{∗} topology and Orb({p}_{n})\to Supp(\mu ) in Hausdorff metric.
The following was proved by Mañé [17]. Denote by \mathcal{M}(f{}_{\mathrm{\Lambda}}) the set of invariant probabilities on the Borel σalgebra of Λ endowed with the weak^{∗} topology.
Lemma 2.14 Let \mathrm{\Lambda}\subset M be a closed finvariant set of f and E\subset {T}_{\mathrm{\Lambda}}M be a continuous invariant subbundle. If there is m>0 such that
for every ergodic \mu \in \mathcal{M}({f}^{m}{}_{\mathrm{\Lambda}}), then E is contracting.
Proof of Theorem 1.4 Let f\in {\mathcal{G}}_{4}\cap {\mathcal{G}}_{5}\cap {\mathcal{G}}_{6} have the ergodic shadowing property in a locally maximal Λ. Then by Proposition 2.11, we know that Λ admits a dominated splitting {T}_{\mathrm{\Lambda}}M=E\oplus F. Since f has the ergodic shadowing property in Λ, by Lemma 2.1, Remark 2.12, and Lemma 2.13, there is a sequence of periodic points such that Orb({p}_{n})\to Supp(\mu )=\mathrm{\Lambda} in the Hausdorff metric. By Proposition 2.11, we have
By Lemma 2.14, E is contracting. Similarly, we can show that F is expanding. □
Corollary 2.15 For {C}^{1}generic f, if f has the ergodic shadowing property, then f is transitive Anosov.
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Acknowledgements
This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2014R1A1A1A05002124).
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Lee, M. The ergodic shadowing property from the robust and generic view point. Adv Differ Equ 2014, 170 (2014). https://doi.org/10.1186/168718472014170
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DOI: https://doi.org/10.1186/168718472014170
Keywords
 ergodic shadowing
 shadowing
 locally maximal
 generic
 Anosov