Volumepreserving diffeomorphisms with inverse shadowing
 Manseob Lee^{1}Email author
https://doi.org/10.1186/1029242X2012275
© Lee; licensee Springer 2012
Received: 21 May 2012
Accepted: 13 November 2012
Published: 28 November 2012
Abstract
Let f be a volumepreserving diffeomorphism of a closed ${C}^{\mathrm{\infty}}$ ndimensional Riemannian manifold M. In this paper, we prove the equivalence between the following conditions:
(a) f belongs to the ${C}^{1}$interior of the set of volumepreserving diffeomorphisms which satisfy the inverse shadowing property with respect to the continuous methods,
(b) f belongs to the ${C}^{1}$interior of the set of volumepreserving diffeomorphisms which satisfy the weak inverse shadowing property with respect to the continuous methods,
(c) f belongs to the ${C}^{1}$interior of the set of volumepreserving diffeomorphisms which satisfy the orbital inverse shadowing property with respect to the continuous methods,
(d) f is Anosov.
MSC:37C50, 34D10.
Keywords
1 Introduction
Let M be a closed ${C}^{\mathrm{\infty}}$ ndimensional Riemannian 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 $f:M\to M$ be a diffeomorphism, and let $\mathrm{\Lambda}\subset M$ 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 any $\u03f5>0$, there is $\delta >0$ such that for any δpseudo orbit ${\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda}$ of f, there is $y\in M$ such that $d({f}^{i}(y),{x}_{i})<\u03f5$ for $i\in \mathbb{Z}$. Note that in this definition, the shadowing point $y\in M$ is not necessarily contained in Λ. We say that f belongs to the ${C}^{1}$interior shadowing property if there is a ${C}^{1}$neighborhood $\mathcal{U}(f)$ of f such that for any $g\in \mathcal{U}(f)$, g has the shadowing property.
The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory [1].
Now, we introduce the notion of the inverse shadowing property which is a ‘dual’ notion of the shadowing property. The inverse shadowing property was introduced by Corless and Pilyugin in [2], and the qualitative theory of dynamical systems with the property was developed by various authors (see [2–7]). In this paper, we introduce the various inverse shadowing properties.
Note that if f has the inverse shadowing property with respect to the class ${\mathcal{T}}_{\alpha}(f)$ ($\alpha =0,c,h,d$), then it has the orbital inverse shadowing property with respect to the class ${\mathcal{T}}_{\alpha}(f)$ ($\alpha =0,c,h,d$). But the converse does not hold. Indeed, an irrational rotation on the unit circle has the orbital inverse shadowing property but does not have the inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$. We say that f belongs to the ${C}^{1}$interior inverse (weak inverse or orbital inverse) shadowing property with respect to the class ${\mathcal{T}}_{\alpha}(f)$, $\alpha =0,c,h,d$ if there is a ${C}^{1}$neighborhood $\mathcal{U}(f)$ of f such that for any $g\in \mathcal{U}(f)$, g has the inverse (weak inverse or orbital inverse) shadowing property with respect to the class ${\mathcal{T}}_{\alpha}(f)$, $\alpha =0,c,h,d$.
Lee [6], showed that a diffeomorphism belongs to the ${C}^{1}$interior inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$ if and only if it is structurally stable. And Pilyugin [7] proved that a diffeomorphism belongs to the ${C}^{1}$interior inverse shadowing property with respect to the class ${\mathcal{T}}_{c}(f)$ if and only if it is structurally stable. Thus, we can restate the above facts as follows.
Theorem 1.1 Let $f\in Diff(M)$. A diffeomorphism f belongs to the ${C}^{1}$interior inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$ (resp. ${\mathcal{T}}_{c}(f)$) if and only if it is structurally stable.
In [3] Choi, Lee and Zhang showed that a diffeomorphism belongs to the ${C}^{1}$interior weak inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$ if and only if it satisfies both Axiom A and the nocycle condition. Moreover, they proved that a diffeomorphism belongs to the ${C}^{1}$interior orbital inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$ if and only if it satisfies both Axiom A and the strong transversal condition. From the above facts, we get the following.
Theorem 1.2 Let $f\in Diff(M)$. If a diffeomorphism f belongs to the ${C}^{1}$interior weak inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$, then f satisfies both Axiom A and the nocycle condition. Moreover, if f belongs to the ${C}^{1}$interior orbital inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$, then it is structurally stable.
By the theorem, even though a diffeomorphism is contained in the ${C}^{1}$interior of the set of diffeomorphisms possessing the weak inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$, it does not necessarily satisfy the strong transversality condition.
A periodic point p of f is hyperbolic if $D{f}^{\pi (p)}$ has eigenvalues with absolute values different from the one, where $\pi (p)$ is the period of p. 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, where $P(g)$ is the set of periodic points of g. It is proved by Hayashi [8] that $f\in \mathcal{F}(M)$ if and only if f satisfies both Axiom A and the nocycle condition.
for all $x\in \mathrm{\Lambda}$ and $n\ge 0$. If $\mathrm{\Lambda}=M$, then we say that f is an Anosov diffeomorphism.
2 Statement of the results
A fundamental problem in differentiable dynamical systems is to understand how a robust dynamic property on the underlying manifold would influence the behavior of the tangent map on the tangent bundle. For instance, in [9], Mañé proved that any ${C}^{1}$ structurally stable diffeomorphism is an Axiom A diffeomorphism. And in [10], Palis extended this result to Ωstable diffeomorphisms.
Let M be a compact ${C}^{\mathrm{\infty}}$ ndimensional Riemannian manifold endowed with a volume form ω. Let μ denote the measure associated with ω that we call the Lebesgue measure, and let d denote the metric induced by the Riemannian structure. Denote by ${Diff}_{\mu}(M)$ the set of diffeomorphisms which preserves the Lebesgue measure μ endowed with the ${C}^{1}$topology. In the volumepreserving case, the Axiom A condition is equivalent to the diffeomorphism being Anosov, since $\mathrm{\Omega}(f)=M$ by the Poincaré recurrence theorem.
We define the set ${\mathcal{F}}_{\mu}(M)$ as the set of diffeomorphisms $f\in {Diff}_{\mu}(M)$ which has a ${C}^{1}$neighborhood $\mathcal{U}(f)\subset {Diff}_{\mu}(M)$ such that for any $g\in \mathcal{U}(f)$, every periodic point of g is hyperbolic. Note that ${\mathcal{F}}_{\mu}(M)\subset \mathcal{F}(M)$ (see [[11], Corollary 1.2]).
Very recently, Arbieto and Catalan [11] proved that if a volumepreserving diffeomorphism is contained in ${\mathcal{F}}_{\mu}(M)$, then it is Anosov. Indeed, at first they used the Mañé results ([[9], Proposition II.1]). Then they showed that $\overline{P(f)}$ is hyperbolic, where $P(f)$ is the set of periodic points of f. And they proved that the nonwandering set $\mathrm{\Omega}(f)=\overline{P(f)}$ by Pugh’s closing lemma. Finally, by the Poincaré recurrence theorem, $\mathrm{\Omega}(f)=M$. From the above facts, we can restate the theorem as follows.
Theorem 2.1 Any diffeomorphism in ${\mathcal{F}}_{\mu}(M)$ is Anosov.
In [12], Lee showed that if a volumepreserving diffeomorphism belongs to the ${C}^{1}$interior expansive or ${C}^{1}$interior shadowing property, then it is Anosov. And [13] proved that if a volumepreserving diffeomorphism belongs to the ${C}^{1}$interior weak shadowing property or ${C}^{1}$interior weak limit shadowing property, then it is Anosov. From these results, we study the cases when a volumepreserving diffeomorphism f is in ${C}^{1}$interior various inverse shadowing property with respect to the class ${\mathcal{T}}_{d}(f)$, then it is Anosov. Let $int{\mathcal{IS}}_{\mu}(M)$ denote the set of volumepreserving diffeomorphisms in ${Diff}_{\mu}(M)$ satisfying the inverse shadowing property with respect to the class ${\mathcal{T}}_{d}$, and let $int{\mathcal{WIS}}_{\mu}(M)$ (respect., $int{\mathcal{OIS}}_{\mu}(M)$) denote the set of volumepreserving diffeomorphisms in ${Diff}_{\mu}(M)$ satisfying the weak inverse shadowing property with respect to the class ${\mathcal{T}}_{d}$ (respect., the orbital inverse shadowing property with respect to the class ${\mathcal{T}}_{d}$). From now, we only consider the class ${\mathcal{T}}_{d}$ when we mention the inverse shadowing property; that is, the ‘inverse shadowing property’ implies the ‘inverse shadowing property with respect to the class ${\mathcal{T}}_{d}$’. Now we are in a position to state the theorem of our paper.
where ${\mathcal{AN}}_{\mu}(M)$ is the set of Anosov volumepreserving diffeomorphisms in ${Diff}_{\mu}(M)$.
3 Proof of Theorem 2.2
Let M be a compact ${C}^{\mathrm{\infty}}$ ndimensional Riemannian manifold endowed with a volume form ω, and let $f\in {Diff}_{\mu}(M)$. To prove the results, we will give the following wellknown Franks lemma for the conservative case, stated and proved in [[14], Proposition 7.4].
Lemma 3.1 Let $f\in {Diff}_{\mu}^{1}(M)$ and $\mathcal{U}$ be a ${C}^{1}$neighborhood of f in ${Diff}_{\mu}^{1}(M)$. Then there exist a ${C}^{1}$neighborhood ${\mathcal{U}}_{0}\subset \mathcal{U}$ of f and $\u03f5>0$ such that if $g\in {\mathcal{U}}_{0}$, for any finite finvariant set $E=\{{x}_{1},\dots ,{x}_{m}\}$, any neighborhood U of E and any volumepreserving linear maps ${L}_{j}:{T}_{{x}_{j}}M\to {T}_{g({x}_{j})}M$ with $\parallel {L}_{j}{D}_{{x}_{j}}g\parallel \le \u03f5$ for all $j=1,\dots ,m$, there is a conservative diffeomorphism ${g}_{1}\in \mathcal{U}$ coinciding with f on E and out of U, and ${D}_{{x}_{j}}{g}_{1}={L}_{j}$ for all $j=1,\dots ,m$.
Remark 3.2 From the Moser theorem (see [15]), there is a smooth conservative change of coordinates ${\phi}_{x}:U(x)\to {T}_{x}M$ such that , where $U(x)$ is a small neighborhood of $x\in M$.
Proposition 3.3 If $f\in int{\mathcal{IS}}_{\mu}(M)$, then every periodic point of f is hyperbolic.
Proof Take $f\in int{\mathcal{IS}}_{\mu}(M)$ and $\mathcal{U}(f)$ is a ${C}^{1}$neighborhood of $f\in int{\mathcal{IS}}_{\mu}(M)$. Let $\u03f5>0$ and $\mathcal{V}(f)\subset {\mathcal{U}}_{0}(f)$ be a corresponding number and a ${C}^{1}$neighborhood given by Lemma 3.1. Suppose that there exists a nonhyperbolic periodic point $p\in P(g)$ for some $g\in \mathcal{V}(f)$. To simplify the notation of the proof, we may assume that $g(p)=p$. Then there is at least one eigenvalue λ of ${D}_{p}g$ such that $\lambda =1$. Denote by ${E}_{p}^{c}$ the eigenspace corresponding to λ. Then we see that if $\lambda \in \mathbb{R}$, then $dim{E}_{p}^{c}=1$, and if $\lambda \in \mathbb{C}$, then $dim{E}_{p}^{c}=2$.
Thus, ${g}_{1}$ does not have the inverse shadowing property.
This is a contradiction.
where ${y}^{\mathrm{\prime}k}={\phi}_{p}^{1}({A}^{k}({w}_{1},\dots ,{w}_{n}))$. This is a contradiction since $f\in int{\mathcal{IS}}_{\mu}(M)$.
Then ${\mathcal{L}}_{p}$ is an arc such that

${g}_{1}^{i}({\mathcal{L}}_{p})\cap {g}_{1}^{j}({\mathcal{L}}_{p})=\mathrm{\varnothing}$ for $0\le i\ne j\le l1$,

${g}_{1}^{l}({\mathcal{L}}_{p})={\mathcal{L}}_{p}$, and

${g}_{1}^{l}{}_{{\mathcal{L}}_{p}}$ is the identity map.
Note that ${g}_{1}$ has the inverse shadowing property if and only if ${g}_{1}^{k}$ has the inverse shadowing property for all $k\in \mathbb{Z}$ (see [6]). As in the first case, we can show that ${g}_{1}$ does not have the inverse shadowing property, which contradicts the fact that ${g}_{1}\in \mathcal{U}(f)$. Thus, every periodic point of $f\in int{\mathcal{IS}}_{\mu}(M)$ is hyperbolic. □
Proposition 3.4 If $f\in int{\mathcal{WIS}}_{\mu}(M)$, then every periodic point of f is hyperbolic.
This is a contradiction.
Thus, ${g}_{1}$ does not have the weak inverse shadowing property. This is a contradiction.
If $\lambda \in \mathbb{C}$, then as in the proof of Proposition 3.3, for ${g}_{1}\in \mathcal{V}(f)$, we can take $l>0$ such that ${D}_{p}{g}_{1}^{l}(v)=v$ for any $v\in {\phi}_{p}({B}_{\alpha}(p))\subset {T}_{p}M$. As in the previous argument, we get a contradiction. Thus, every periodic point of $f\in int{\mathcal{WIS}}_{\mu}(M)$ is hyperbolic. Consequently, if $f\in int{\mathcal{WIS}}_{\mu}(M)$, then $f\in {\mathcal{F}}_{\mu}(M)$. □
Proposition 3.5 If $f\in int{\mathcal{OIS}}_{\mu}(M)$, then every periodic point of f is hyperbolic.
This is a contradiction since $f\in int{\mathcal{OIS}}_{\mu}(M)$.
This is a contradiction.
Thus, ${g}_{1}$ does not have the orbital inverse shadowing property. This is a contradiction.
If $\lambda \in \mathbb{C}$, then as in the proof of Proposition 3.4, for ${g}_{1}\in \mathcal{V}(f)$, we can take $l>0$ such that ${D}_{p}{g}_{1}^{l}(v)=v$ for any $v\in {\phi}_{p}({B}_{\alpha}(p))\subset {T}_{p}M$. As in the previous argument, in order to reach the same contradiction. Thus, every periodic point of $f\in int{\mathcal{OIS}}_{\mu}(M)$ is hyperbolic. □
Declarations
Acknowledgements
The author wishes to express their appreciation to the referee for their careful reading of the manuscript and valuable suggestions. This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology, Korea (No. 20110007649).
Authors’ Affiliations
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