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Generic diffeomorphisms with weak limit shadowing
Advances in Difference Equations volume 2013, Article number: 27 (2013)
Abstract
In this paper, we show that if a ${C}^{1}$generic diffeomorphism has the weak limit shadowing property on the chain recurrent set, then the diffemorphism satisfies Axiom A and the nocycle condition.
MSC:37C50, 37D30.
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 $f\in Diff(M)$ and Λ be a closed finvariant set. 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 $a\le i\le b1$.
We say that f has the shadowing property on 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 $y\in M$ such that $d({f}^{i}(y),{x}_{i})<\u03f5$ for all $a\le i\le b1$. In the dynamical systems, the shadowing theory is a very useful notion. In fact, it deals with the stability theorem (see [1]). For instance, Robinson [2] proved that if a diffeomorphism f is structurally stable, then it has the shadowing property. In [3] Sakai showed that f belongs to the ${C}^{1}$interior of the shadowing property if and only if f is structurally stable. In this paper, we deal with another shadowing property, that is, the weak limit shadowing property which was studied by [4].
We say that f has the weak limit shadowing property on Λ (or Λ is weak limit shadowable for f) if there exists a $\delta >0$ with the following property: if a sequence ${\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda}$ is a δpseudo orbit of f, for which relations $d(f({x}_{i}),{x}_{i+1})\to 0$ as $i\to +\mathrm{\infty}$ and $d({f}^{1}({x}_{i+1}),{x}_{i})\to 0$ as $i\to \mathrm{\infty}$ hold, then there is a point $y\in M$ such that $d({f}^{i}(y),{x}_{i})\to 0$ as $i\to \pm \mathrm{\infty}$. Note that if f has the limit shadowing property, then f has the weak limit shadowing property. But the converse is not true (see [[4], Example 4]). Denote by $P(f)$ the set of periodic points of f. Then $P(f)\subset \mathrm{\Omega}(f)\subset \mathcal{R}(f)$, where $\mathrm{\Omega}(f)$ is the set of nonwandering points of f, and $\mathcal{R}(f)$ is the set of chain recurrent points of f. Note that if f satisfies Axiom A and the nocycle condition, then $\mathrm{\Omega}(f)=\mathcal{R}(f)$. We say that f has the slimit shadowing property on Λ if for any $\u03f5>0$, there is a $\delta >0$ such that for any δlimit pseudo orbit $\xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda}$, there is a point $y\in M$ such that $d({f}^{i}(y),{x}_{i})<\u03f5$ for all $i\in \mathbb{Z}$, and $d({f}^{i}(y),{x}_{i})\to 0$ as $i\to \pm \mathrm{\infty}$. Clearly, the weak limit shadowing property is a weak notion of the slimit shadowing property. 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. Very recently, Sakai [5] showed that if a ${C}^{1}$generic diffeomorphism f has the slimit shadowing property on $\mathcal{R}(f)$, then f satisfies Axiom A and the nocycle condition. The result is motivation for this study. The main theorem of the paper is as follows.
Theorem 1.1 For ${C}^{1}$generic f, if f has the weak limit shadowing property on $\mathcal{R}(f)$, then f satisfies Axiom A and the nocycle condition.
2 Proof of Theorem 1.1
Let M be as before and $f\in Diff(M)$. Let $p\in P(f)$ be a hyperbolic saddle with period $\pi (p)>0$. The stable manifold ${W}^{s}(p)$ and the unstable manifold ${W}^{u}(p)$ are defined as follows. It is well known that if p is a hyperbolic periodic point of f with a period k, then the sets
are ${C}^{1}$injectively immersed submanifolds of M. Let $p,q\in P(f)$ be saddles. Let $P(f)$ be the set of periodic points of f. Denote by ${\mathcal{O}}_{f}(p)$ the periodic forbit of $p\in P(f)$. We denote $p\sim q$ if the intersections ${W}^{s}({\mathcal{O}}_{f}(p))\u22d4{W}^{u}({\mathcal{O}}_{f}(q))\ne \mathrm{\varnothing}$ and ${W}^{u}({\mathcal{O}}_{f}(p))\u22d4{W}^{s}({\mathcal{O}}_{f}(q))\ne \mathrm{\varnothing}$. Then we know that if $p\sim q$, then $index(p)=index(q)$. Here $index(p)$ is the dimension of the stable manifold of p, that is, $dim{W}^{s}(p)$.
Proposition 2.1 There is a residual set ${\mathcal{G}}_{1}\subset Diff(M)$ such that for any $f\in \mathcal{G}$, if $f{}_{\mathcal{R}(f)}$ has the weak limit shadowing property, then for any saddles $p,q\in P(f)$, $index(p)=index(q)$.
To prove Proposition 2.1, we need the following lemma.
Lemma 2.2 Let $p,q\in P(f)$ be saddles. If f has the weak limit shadowing property on $\mathcal{R}(f)$, then ${W}^{s}({\mathcal{O}}_{f}(p))\cap {W}^{u}({\mathcal{O}}_{f}(q))\ne \mathrm{\varnothing}$.
Proof Suppose that f has the weak limit shadowing property on $\mathcal{R}(f)$. For any saddles $p,q\in P(f)$, we show that ${W}^{u}({\mathcal{O}}_{f}(p))\cap {W}^{s}({\mathcal{O}}_{f}(q))\ne \mathrm{\varnothing}$. For the sake of simplicity, we may assume that $f(p)=p$ and $f(q)=q$. Let $\delta >0$ be the number of the weak limit shadowing property of f such that $d(p,q)<\delta $. We construct δlimit pseudo orbit $\xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathcal{R}(f)$ as follows. (i) ${x}_{0}=p$, (ii) ${x}_{i}={f}^{i}(p)$ for all $i>0$, (iii) ${x}_{i}={f}^{i}(q)$ for all $i\ge 1$. Then the δlimit pseudo orbit
and it is clear that $\xi \subset \mathcal{R}(f)$. Since f has the weak shadowing property on $\mathcal{R}(f)$, there is a point $y\in M$ such that $d({f}^{i}(y),{x}_{i})\to 0$ as $i\to \pm \mathrm{\infty}$. Then ${f}^{i}(y)\to p$ as $i\to \mathrm{\infty}$ and ${f}^{1+i}(y)\to q$ as $i\to \mathrm{\infty}$. Hence, $y\in {W}^{s}(p)$ and $f(y)\in {W}^{s}(q)$. Thus, ${W}^{u}(p)\cap {W}^{s}(q)\ne \mathrm{\varnothing}$. □
The following is called the KupkaSmale theorem.
Lemma 2.3 There is a residual set ${\mathcal{G}}_{1}\subset Diff(M)$ such that for any $f\in \mathcal{G}$, every periodic point is hyperbolic, and the stable manifolds and the unstable manifolds of periodic points are all transverse.
Proof of Proposition 2.1 Let $f\in {\mathcal{G}}_{1}$, and let $p,q\in P(f)$ be saddles. Suppose that f has the weak limit shadowing property on $\mathcal{R}(f)$. Let $\delta >0$ be the number of the weak limit shadowing property of f such that $d(p,q)<\delta $. Then we will drive a contradiction, we may assume that $index(p)\ne index(q)$. Then we know that $dim{W}^{s}(p)+dim{W}^{u}(q)<dimM$ or $dim{W}^{u}(p)+dim{W}^{s}(q)<dimM$. In this proof, we consider that $dim{W}^{s}(p)+dim{W}^{u}(q)<dimM$ (the other case is similar). Since $f\in {\mathcal{G}}_{1}$, ${W}^{s}(p)\cap {W}^{u}(q)=\mathrm{\varnothing}$. Since f has the weak limit shadowing property on $\mathcal{R}(f)$, by Lemma 2.2, ${W}^{s}(p)\cap {W}^{u}(q)\ne \mathrm{\varnothing}$. This is a contradiction. □
Let $p\in P(f)$ be a hyperbolic saddle with a period $\pi (p)>0$. Then there are the local stable manifold ${W}_{\u03f5}^{s}(p)$ and the 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 following lemma shows that if f has the slimit shadowing property on $\mathcal{R}(f)$, then the numbers of sinks and sources are finite (see [[5], Lemma 2]). From the above facts, we show that if f has the weak limit shadowing property on $\mathcal{R}(f)$, then the numbers of sinks and sources are finite.
Lemma 2.4 Let f have the weak limit shadowing property on $\mathcal{R}(f)$, and let $\delta >0$ be the number of the weak limit shadowing property of f. For any saddle $q\in P(f)$, if $p\in P(f)$ is a sink or a source, then $d(p,q)\ge \delta $.
Proof We will derive a contradiction. Suppose that $q\in P(f)$ is a saddle and $p\in P(f)$ is a sink with $d(p,q)<\delta $. For the sake of simplicity, we may assume that $f(p)=p$, $f(q)=q$. Since q is a saddle, there is $\u03f5(q)>0$ such that if for any $x\in M$, $d({f}^{i}(x),{f}^{i}(q))\le \u03f5(q)$ as $i\to \mathrm{\infty}$, then $x\in {W}_{\u03f5(q)}^{s}(q)$, and if $x\in M$, $d({f}^{i}(x),{f}^{i}(q))\le \u03f5(q)$ as $i\to \mathrm{\infty}$, then $x\in {W}_{\u03f5(q)}^{u}(q)$. Then we may assume that $d(p,q)>\u03f5(q)$. Then we construct a δlimit pseudo orbit $\xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathcal{R}(f)$ as follows. Put ${x}_{i}={f}^{i}(p)$ for $i\ge 0$ and ${x}_{i}={f}^{i}(q)$ for $i\ge 1$. Then $\xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}$ is clearly a δlimit pseudo orbit of f, and $\xi ={\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathcal{R}(f)$. Since f has the weak limit shadowing property on $\mathcal{R}(f)$, there is a point $y\in M$ such that $d({f}^{i}(y),{x}_{i})\to 0$ as $i\to \pm \mathrm{\infty}$. Since p is a sink, $d({f}^{i}(y),{x}_{i})=d({f}^{i}(y),p)\to 0$ as $i\to \mathrm{\infty}$. Then $y=p$. Since $d({f}^{i}(y),{x}_{i})=d({f}^{i}(y),q)\to 0$ as $i\to \mathrm{\infty}$, there is $k>0$ such that $d({f}^{k+i}(y),{f}^{k+i}(q))=d({f}^{k+i}(y),q)\le \u03f5(q)$ for $i\ge 0$. Then ${f}^{k}(y)\in {W}_{\u03f5(q)}^{s}(q)$. Since $y=p$, we know that $d(p,q)\le \u03f5(q)$. This is a contradiction. □
Let p be a periodic point of f, and let $0<\delta <1$. We say p has a δweak eigenvalue provided ${D}_{p}{f}^{\pi (p)}$ has an eigenvalue λ such that ${(1\delta )}^{\pi (p)}<\lambda <{(1+\delta )}^{\pi (p)}$. We say that the periodic point has a real spectrum if all of its eigenvalues are real and a simple spectrum if all of its eigenvalues have multiplicity one. The following lemma will play a crucial role in our proof.
Lemma 2.5 [[6], Lemma 5.1]
There is a residual set ${\mathcal{G}}_{2}\subset Diff(M)$ such that for any $f\in {\mathcal{G}}_{2}$,

(a)
for any $\eta >0$, if for any ${C}^{1}$neighborhood $\mathcal{U}(f)$ of f, there exist $g\in \mathcal{U}(f)$ and ${p}_{g},{q}_{g}\in P(g)$ with the same period such that $d({p}_{g},{q}_{g})<\eta $, then there exist $p,q\in P(f)$ with the same period such that $d(p,q)<\eta $;

(b)
for any $\eta >0$, if for any ${C}^{1}$neighborhood $\mathcal{U}(f)$ of f, there exist $g\in \mathcal{U}(f)$ and ${p}_{g}\in P(g)$ with an ηweak eigenvalue, then there exist $p\in P(f)$ with a 2ηweak eigenvalue;

(c)
for any $\eta >0$, if $q\in P(f)$ with an ηweak eigenvalue and a real spectrum, then there exists $p\in P(f)$ with an ηweak eigenvalue with a simple real spectrum.
Lemma 2.6 [[7], 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 $\eta >0$, if for any ${C}^{1}$neighborhood $\mathcal{U}(f)$ of f, there exist $g\in \mathcal{U}(f)$ and ${p}_{g},{q}_{g}\in P(g)$ with the same period such that $d({p}_{g},{q}_{g})<\eta $ with different indices, then there exist $p,q\in P(f)$ with the same period such that $d(p,q)<\eta $ with different indices.
The following socalled Franks lemma will play an essential role in our proof.
Lemma 2.7 Let $\mathcal{U}(f)$ be any given ${C}^{1}$neighborhood of f. Then there exists $\u03f5>0$ and a ${C}^{1}$neighborhood ${\mathcal{U}}_{0}(f)\subset \mathcal{U}(f)$ of f such that for given $g\in {\mathcal{U}}_{0}(f)$, a finite set $\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}$, a neighborhood U of $\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}$ and linear maps ${L}_{i}:{T}_{{x}_{i}}M\to {T}_{g({x}_{i})}M$ satisfying $\parallel {L}_{i}{D}_{{x}_{i}}g\parallel \le \u03f5$ for all $1\le i\le N$, there exists ${g}^{\prime}\in \mathcal{U}(f)$ such that ${g}^{\prime}(x)=g(x)$ if $x\in \{{x}_{1},{x}_{2},\dots ,{x}_{N}\}\cup (M\setminus U)$ and ${D}_{{x}_{i}}{g}^{\prime}={L}_{i}$ for all $1\le i\le N$.
If $p\in P(f)$ is hyperbolic, then for any $g\in \mathcal{U}(f)$, there is a unique hyperbolic periodic point ${p}_{g}\in P(g)$ nearby p such that $\pi ({p}_{g})=\pi (p)$ and $index({p}_{g})=index(p)$, where $index=dim{W}^{s}(p)$. Such a ${p}_{g}$ is called the continuation of p.
Lemma 2.8 There is a residual set ${\mathcal{G}}_{4}\subset Diff(M)$ such that for any $f\in {\mathcal{G}}_{4}$, if f has the weak limit shadowing property on $\mathcal{R}(f)$, then there is $\eta >0$ such that f has no ηweak eigenvalue.
Proof Let $f\in {\mathcal{G}}_{4}={\mathcal{G}}_{2}\cap {\mathcal{G}}_{3}$. To derive a contradiction, we may assume that for any $\eta >0$, there is a hyperbolic periodic point ${q}_{g}$ of g (${C}^{1}$nearby f) such that ${q}_{g}$ has an ηweak eigenvalue and a simple real spectrum. Let $\delta >0$ be the number of the weak limit shadowing property of f such that $0<\eta <\delta /2$. For the sake of simplicity, we assume that ${q}_{g}$ is a fixed point. By Lemma 2.7, there is h ${C}^{1}$close to g and h ${C}^{1}$nearby f such that ${q}_{h}$ has 1 as an eigenvalue. By Lemma 2.7 and as in the proof of [[8], Lemma 2.4], we can construct an ${h}^{l}$ ($l>0$)invariant small arc ${\mathcal{I}}_{{q}_{h}}$ of ${h}^{l}$ containing ${q}_{h}$ such that $d({p}_{h},{r}_{h})<\eta $, where ${p}_{h}$, ${r}_{h}$ are the end points of ${\mathcal{I}}_{{q}_{h}}$, ${h}^{l}({p}_{h})={p}_{h}$, ${h}^{l}({r}_{h})={r}_{h}$ and ${p}_{h}$, ${r}_{h}$ are hyperbolic saddles and different indices. Since $f\in {\mathcal{G}}_{4}$, there exist $p,r\in P(f)$ with the same period such that $d(p,r)<\eta $ with different indices. Since f has the weak limit shadowing property on $\mathcal{R}(f)$, and by Lemma 2.4, $p,r\in P(f)$ are saddles. Since $d(p,r)<\delta $, by Proposition 2.1, we know that $index(p)=index(r)$. But, since $index(p)\ne index(r)$, this is a contradiction. □
Denote by $\mathcal{F}(M)$ the ${C}^{1}$interior of the set of diffeomorphisms of M whose periodic points are all hyperbolic. In [9], Hayashi showed that if $f\in \mathcal{F}(M)$, then f satisfies Axiom A and the nocycle condition. To prove Theorem 1.1, it is enough to show $f\in \mathcal{F}(M)$.
End of proof of Theorem 1.1 Let $f\in {\mathcal{G}}_{4}$, and let f have the weak limit shadowing property on $\mathcal{R}(f)$. If not, then $f\notin \mathcal{F}(M)$. There is g ${C}^{1}$closed to f and a nonhyperbolic periodic point ${p}_{g}$ such that the point ${p}_{g}$ has an $\eta /2$ weak eigenvalue. Since $f\in {\mathcal{G}}_{4}$, there is $p\in P(f)$ such that p has an ηweak eigenvalue. By Lemma 2.8, this is a contradiction. □
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Acknowledgements
We wish to thank the referee for carefully reading of the manuscript and for providing us with many good suggestions. GL is supported by the Doctoral Science Foundation of Liaoning Province by Hall of Liaoning province science and technology (No. 20121055). KL is supported by the National Research Foundation (NRF) of Korea funded by the Korean Government (No. 20110015193). ML is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology, Korea (No. 20110007649).
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Lu, G., Lee, K. & Lee, M. Generic diffeomorphisms with weak limit shadowing. Adv Differ Equ 2013, 27 (2013). https://doi.org/10.1186/16871847201327
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Keywords
 shadowing
 weak limit shadowing
 Ωstable
 generic