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Volume preserving diffeomorphisms with orbital shadowing
Journal of Inequalities and Applications volumeĀ 2013, ArticleĀ number:Ā 18 (2013)
Abstract
Let f be a volumepreserving diffeomorphism of a closed {C}^{\mathrm{\u0101\x88\x9e}} Riemannian manifoldĀ M. In this paper, we show that the following are equivalent:
(a) f belongs to the {C}^{1}interior of the set of volumepreserving diffeomorphisms with orbital shadowing,
(b) f is Anosov.
MSC:37C50, 34D10.
1 Introduction
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 [1], MaĆ±Ć© proved that any {C}^{1} structurally stable diffeomorphism is an Axiom A diffeomorphism. And in [2], Palis extended this result to Ī©stable diffeomorphisms. Let M be a closed {C}^{\mathrm{\u0101\x88\x9e}} Riemannian manifold endowed with a volume form Ļ. Let Ī¼ denote the Lebesgue measure associated to Ļ, and let d denote the metric induced on M by the Riemannian structure. Denote by {Diff}_{\mathrm{\u012a\xbc}}(M) the set of diffeomorphisms which preserves the Lebesgue measure Ī¼ endowed with the {C}^{1}topology. We know that every volume preserving diffeomorphism satisfying Axiom A is Anosov (for more details, see [3]).
For \mathrm{\u012a\u201c}>0, a sequence of points {\{{x}_{i}\}}_{i=a}^{b} (\u0101\x88\x92\mathrm{\u0101\x88\x9e}\u0101\x89\xa4a<b\u0101\x89\xa4\mathrm{\u0101\x88\x9e}) in M is called a Ī“pseudoorbit of f if d(f({x}_{i}),{x}_{i+1})<\mathrm{\u012a\u201c} for all a\u0101\x89\xa4i\u0101\x89\xa4b\u0101\x88\x921. Let \mathrm{\u012a\x9b}\u0101\x8a\x82M be a closed finvariant set. We say that f has the shadowing property on Ī (or Ī is shadowable) if for every \mathrm{\u013b\mu}>0, there is \mathrm{\u012a\u201c}>0 such that for any Ī“pseudoorbit {\{{x}_{i}\}}_{i=a}^{b}\u0101\x8a\x82\mathrm{\u012a\x9b} of f (\u0101\x88\x92\mathrm{\u0101\x88\x9e}\u0101\x89\xa4a<b\u0101\x89\xa4\mathrm{\u0101\x88\x9e}), there is a point y\u0101\x88\x88M such that d({f}^{i}(y),{x}_{i})<\mathrm{\u013b\mu} for all a\u0101\x89\xa4i\u0101\x89\xa4b\u0101\x88\x921. It is easy to see that f has the shadowing property on Ī if and only if {f}^{n} has the shadowing property on Ī for n\u0101\x88\x88\mathbb{Z}\u0101\x88\x96\{0\}. The notion of pseudoorbits often appears in several methods of the modern theory of dynamical systems. Moreover, the shadowing property plays an important role in the investigation of stability theory. In fact, Pilyugin [4] and Robinson [5] showed that if a diffeomorphism f is structurally stable, then f has the shadowing property. Moreover, Sakai [6] proved that if there is a {C}^{1}neighborhood \mathcal{U}(f) of f such that for any g\u0101\x88\x88\mathcal{U}(f), g has the shadowing property, then f is structurally stable. For each x\u0101\x88\x88M, let {\mathcal{O}}_{f}(x) be the orbit of f through x; that is,
We say that f has the orbital shadowing property on Ī (or Ī is orbitally shadowable) if for any \mathrm{\u013b\mu}>0, there exists \mathrm{\u012a\u201c}>0 such that for any Ī“pseudoorbit \mathrm{\u012a\xbe}={\{{x}_{i}\}}_{i\u0101\x88\x88\mathbb{Z}}\u0101\x8a\x82\mathrm{\u012a\x9b}, we can find a point y\u0101\x88\x88M such that
where {B}_{\mathrm{\u013b\mu}}(A) denotes the Ļµneighborhood of a set A\u0101\x8a\x82M. f is said to have the weak shadowing property on Ī (or Ī is weakly shadowable) if for any \mathrm{\u013b\mu}>0, there exists \mathrm{\u012a\u201c}>0 such that for any Ī“pseudoorbit \mathrm{\u012a\xbe}={\{{x}_{i}\}}_{i\u0101\x88\x88\mathbb{Z}}\u0101\x8a\x82\mathrm{\u012a\x9b}, there is a point y\u0101\x88\x88M such that \mathrm{\u012a\xbe}\u0101\x8a\x82{B}_{\mathrm{\u013b\mu}}({\mathcal{O}}_{f}(y)). Note that if f has the shadowing property, then f has the orbital shadowing property, but the converse is not true (see [7]). It is easy to see that f has the orbital shadowing property on Ī if and only if {f}^{n} has the orbital shadowing property on Ī for n\u0101\x88\x88\mathbb{Z}\u0101\x88\x96\{0\}.
We say that Ī is hyperbolic if the tangent bundle {T}_{\mathrm{\u012a\x9b}}M has a Dfinvariant splitting {E}^{s}\u0101\x8a\x95{E}^{u} and there exist constants C>0 and 0<\mathrm{\u012a\xbb}<1 such that
for all x\u0101\x88\x88\mathrm{\u012a\x9b} and n\u0101\x89\u201e0.
We denote by {\mathcal{F}}_{\mathrm{\u012a\xbc}}(M) the set of diffeomorphisms f\u0101\x88\x88{Diff}_{\mathrm{\u012a\xbc}}(M) which have a {C}^{1}neighborhood \mathcal{U}(f)\u0101\x8a\x82{Diff}_{\mathrm{\u012a\xbc}}(M) such that for any g\u0101\x88\x88\mathcal{U}(f), every periodic point of g is hyperbolic.
Very recently, Arbieto and Catalan [3] proved that every volume preserving diffeomorphism in {\mathcal{F}}_{\mathrm{\u012a\xbc}}(M) is Anosov. To prove this, they used MaĆ±Ć©ās results in [[1], Proposition II.1] and showed that \stackrel{\u0100\AE}{P(f)} is hyperbolic if f\u0101\x88\x88{\mathcal{F}}_{\mathrm{\u012a\xbc}}(M). Thus, we have the following theorem.
Theorem 1.1 [[3], Theorem 1.1]
Every diffeomorphism in {\mathcal{F}}_{\mathrm{\u012a\xbc}}(M) is Anosov.
Let int{\mathcal{OS}}_{\mathrm{\u012a\xbc}}(M) denote the {C}^{1}interior of the set of volume preserving diffeomorphisms in {Diff}_{\mathrm{\u012a\xbc}}(M) satisfying the orbital shadowing property. In [7], the authors proved that the {C}^{1}interior of the set of diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms. Note that if a diffeomorphism satisfies structurally stable then it is not Anosov in general. But the converse is true. Finally, we prove the following theorem.
Theorem 1.2 The set {\mathcal{AN}}_{\mathrm{\u012a\xbc}}(M) of Anosov diffeomorphisms in {Diff}_{\mathrm{\u012a\xbc}}(M) coincides with the {C}^{1}interior of the set of diffeomorphisms in {Diff}_{\mathrm{\u012a\xbc}}(M) with orbital shadowing; that is, {\mathcal{AN}}_{\mathrm{\u012a\xbc}}(M)=int{\mathcal{OS}}_{\mathrm{\u012a\xbc}}(M).
2 Proof of Theorem 1.2
Remark 2.1 Let f\u0101\x88\x88{Diff}_{\mathrm{\u012a\xbc}}^{1}(M). From Moserās theorem (see [8]), we can find a smooth conservative change of coordinates {\mathrm{\u013b\x86}}_{x}:U(x)\u0101\x86\x92{T}_{x}M such that {\mathrm{\u013b\x86}}_{x}(x)=0, where U(x) is a small neighborhood of x\u0101\x88\x88M.
Recall that f has the orbital shadowing property on Ī (Ī is orbitally shadowable) if for any \mathrm{\u013b\mu}>0, there is \mathrm{\u012a\u201c}>0 such that for any Ī“pseudo orbit \mathrm{\u012a\xbe}={\{{x}_{i}\}}_{i\u0101\x88\x88\mathbb{Z}}\u0101\x8a\x82\mathrm{\u012a\x9b} of f, there is āM such that
Notice that in this definition, only Ī“pseudo orbits of f are contained in Ī, but the shadowing point y\u0101\x88\x88M is not necessarily contained in Ī. To prove our result, we use Franksā lemma which is proved in [[9], Proposition 7.4].
Lemma 2.2 Let f\u0101\x88\x88{Diff}_{\mathrm{\u012a\xbc}}^{1}(M), and \mathcal{U} be a {C}^{1}neighborhood of f in {Diff}_{\mathrm{\u012a\xbc}}^{1}(M). Then there exist a {C}^{1}neighborhood {\mathcal{U}}_{0}\u0101\x8a\x82\mathcal{U} of f and \mathrm{\u013b\mu}>0 such that if g\u0101\x88\x88{\mathcal{U}}_{0}, any finite finvariant set E=\{{x}_{1},\u0101\x80\xa6,{x}_{m}\}, any neighborhood U of E, and any volumepreserving linear maps {L}_{j}:{T}_{{x}_{j}}M\u0101\x86\x92{T}_{g({x}_{j})}M with \u0101\x88\u201e{L}_{j}\u0101\x88\x92{D}_{{x}_{j}}g\u0101\x88\u201e\u0101\x89\xa4\mathrm{\u013b\mu} for all j=1,\u0101\x80\xa6,m, there is a conservative diffeomorphism {g}_{1}\u0101\x88\x88\mathcal{U} coinciding with f on E and out of U, and {D}_{{x}_{j}}{g}_{1}={L}_{j} for all j=1,\u0101\x80\xa6,m.
We introduce the notion of normally hyperbolic which was founded in [10]. Let V\u0101\x8a\x82M be an invariant submanifold of f\u0101\x88\x88{Diff}_{\mathrm{\u012a\xbc}}(M). We say that V is normally hyperbolic if there is a splitting {T}_{V}M=TV\u0101\x8a\x95{N}^{s}\u0101\x8a\x95{N}^{u} such that

the splitting depends continuously on x\u0101\x88\x88V,

{D}_{x}f({N}_{x}^{\mathrm{\u013b\x83}})={N}_{f(x)}^{\mathrm{\u013b\x83}} (\mathrm{\u013b\x83}=s,u) for all x\u0101\x88\x88V,

there are constants C>0 and 0<\mathrm{\u012a\xbb}<1 such that for every unit vector x\u0101\x88\x88{T}_{x}V, {v}^{s}\u0101\x88\x88{N}_{x}^{s} and {v}^{u}\u0101\x88\x88{N}_{x}^{u} (x\u0101\x88\x88V), we have
\u0101\x88\u201e{D}_{x}{f}^{n}\left({v}^{s}\right)\u0101\x88\u201e\u0101\x89\xa4C{\mathrm{\u012a\xbb}}^{n}\u0101\x88\u201e{D}_{x}{f}^{n}(v)\u0101\x88\u201e\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\u0101\x88\u201e{D}_{x}{f}^{n}\left({v}^{u}\right)\u0101\x88\u201e\u0101\x89\u201e{C}^{\u0101\x88\x921}{\mathrm{\u012a\xbb}}^{\u0101\x88\x921}\u0101\x88\u201e{D}_{x}{f}^{n}(v)\u0101\x88\u201e
for all n\u0101\x89\u201e0.
Proposition 2.3 If f\u0101\x88\x88int{\mathcal{OS}}_{\mathrm{\u012a\xbc}}(M), then every periodic point of f is hyperbolic.
Proof Take f\u0101\x88\x88int{\mathcal{OS}}_{\mathrm{\u012a\xbc}}(M), and \mathcal{U}(f) is a {C}^{1}neighborhood of f\u0101\x88\x88{Diff}_{\mathrm{\u012a\xbc}}(M). Let \mathrm{\u013b\mu}>0 and \mathcal{V}(f)\u0101\x8a\x82{\mathcal{U}}_{0}(f) be the number and {C}^{1}neighborhood of f corresponding to \mathcal{U}(f) given by Lemma 2.2. To derive a contradiction, we assume that there exists a nonhyperbolic periodic point p\u0101\x88\x88P(g) for some g\u0101\x88\x88\mathcal{V}(f). To simplify the notation in the proof, we may assume that g(p)=p. Then there is at least one eigenvalue Ī» of {D}_{p}g such that \mathrm{\u012a\xbb}=1.
By making use of Lemma 2.2, we linearize g at p using Moserās theorem; that is, by choosing \mathrm{\u012a\pm}>0 sufficiently small, we construct {g}_{1} {C}^{1}nearby g such that
Then {g}_{1}(p)=g(p)=p.
First, we may assume that \mathrm{\u012a\xbb}\u0101\x88\x88\mathbb{R} with \mathrm{\u012a\xbb}=1. Let v be the associated nonzero eigenvector such that \u0101\x88\u201ev\u0101\x88\u201e=\mathrm{\u012a\pm}/4. Then we can get a small arc
Take \mathrm{\u013b\mu}=\mathrm{\u012a\pm}/8. Let 0<\mathrm{\u012a\u201c}<\mathrm{\u013b\mu} be a number of the orbital shadowing property of {g}_{1} corresponding to Ļµ. Then by our construction of {g}_{1},
Then it is clear that {\mathrm{\u013b\x86}}_{p}^{\u0101\x88\x921}({\mathcal{I}}_{v}) is normally hyperbolic for {g}_{1}. Put {\mathcal{J}}_{p}={\mathrm{\u013b\x86}}_{p}^{\u0101\x88\x921}({\mathcal{I}}_{v}). Given a constant \mathrm{\u012a\u201c}>0, we construct a Ī“pseudo orbit \mathrm{\u012a\xbe}={\{{x}_{i}\}}_{i\u0101\x88\x88\mathbb{Z}}\u0101\x8a\x82{\mathcal{J}}_{p} as follows. For fixed k\u0101\x88\x88\mathbb{Z}, choose distinct points {x}_{0}=p,{x}_{1},{x}_{2},\u0101\x80\xa6,{x}_{k} in {\mathcal{J}}_{p} such that

(a)
d({x}_{i},{x}_{i+1})<\mathrm{\u012a\u201c} for i=0,1,\u0101\x80\xa6,k\u0101\x88\x921,

(b)
d({x}_{\u0101\x88\x92i\u0101\x88\x921},{x}_{\u0101\x88\x92i})<\mathrm{\u012a\u201c} for i=0,\u0101\x80\xa6,k\u0101\x88\x921,

(c)
{x}_{0}=x and d({x}_{\u0101\x88\x92k},{x}_{k})>2\mathrm{\u013b\mu}.
Now, we define \mathrm{\u012a\xbe}={\{{x}_{i}\}}_{i\u0101\x88\x88\mathbb{Z}} by {x}_{ki+j}={x}_{j} for i\u0101\x88\x88\mathbb{Z} and j=\u0101\x88\x92k\u0101\x88\x921,\u0101\x88\x92k\u0101\x88\x922,\u0101\x80\xa6,\u0101\x88\x921,0,1,\u0101\x80\xa6,k\u0101\x88\x921. Since {g}_{1} has the orbital shadowing property, {g}_{1}{}_{{\mathcal{J}}_{p}} must have the orbital shadowing property. Thus, we can find a point y\u0101\x88\x88M such that \mathrm{\u012a\xbe}\u0101\x8a\x82{B}_{\mathrm{\u013b\mu}}({\mathcal{O}}_{{g}_{1}}(y)), and {\mathcal{O}}_{{g}_{1}}(y)\u0101\x8a\x82{B}_{\mathrm{\u013b\mu}}(\mathrm{\u012a\xbe}). For any v\u0101\x88\x88{\mathcal{I}}_{v}, {\mathrm{\u013b\x86}}_{p}^{\u0101\x88\x921}(v)\u0101\x88\x88{\mathcal{J}}_{p}\u0101\x8a\x82{B}_{\mathrm{\u012a\pm}}(p) and
Then {g}_{1}({\mathrm{\u013b\x86}}_{p}^{\u0101\x88\x921}(v))={\mathrm{\u013b\x86}}_{p}^{\u0101\x88\x921}(v). Thus, {g}_{1}^{l}({\mathcal{J}}_{p})={\mathcal{J}}_{p} for some l>0. Now, we show that if {\mathcal{J}}_{p} is normally hyperbolic for {g}_{1}, then the shadowing points belong to {\mathcal{J}}_{p}. Assume that there is a shadowing point y\u0101\x88\x88M\u0101\x88\x96{\mathcal{J}}_{p}. Then by the hyperbolicity, there are l,k\u0101\x88\x88\mathbb{Z} such that d({g}_{1}^{l}(y),{x}_{k})>\mathrm{\u013b\mu}, where {x}_{k}\u0101\x88\x88\mathrm{\u012a\xbe}={\{{x}_{i}\}}_{i\u0101\x88\x88\mathbb{Z}}. This is a contradiction since {g}_{1}{}_{{\mathcal{J}}_{p}} has the orbital shadowing property. Thus, if {\mathcal{J}}_{p} is normally hyperbolic for {g}_{1}, then the shadowing point belongs to {\mathcal{J}}_{p}. Since {g}_{1}{}_{{\mathcal{J}}_{p}} has the orbital shadowing property, from the above facts, we have y\u0101\x88\x88{\mathcal{J}}_{p}. But {g}_{1}^{l}({\mathcal{J}}_{p})={\mathcal{J}}_{p} and so {g}_{1}^{l}{}_{{\mathcal{J}}_{p}} is the identity map. Then {g}_{1}^{l}{}_{{\mathcal{J}}_{p}} does not have the orbital shadowing property. Thus, {g}_{1}{}_{{\mathcal{J}}_{p}} also does not have the orbital shadowing property.
Finally, if \mathrm{\u012a\xbb}\u0101\x88\x88\mathbb{C}, then to avoid the notational complexity, we may assume that g(p)=p. As in the first case, by Lemma 2.2, there are \mathrm{\u012a\pm}>0 and {g}_{1}\u0101\x88\x88\mathcal{V}(f) such that {g}_{1}(p)=g(p)=p and
With a {C}^{1}small modification of the map {D}_{p}g, we may suppose that there is l>0 (the minimum number) such that {D}_{p}{g}^{l}(v)=v for any v\u0101\x88\x88{\mathrm{\u013b\x86}}_{p}({B}_{\mathrm{\u012a\pm}}(p))\u0101\x8a\x82{T}_{p}M. Then we can go on with the previous argument in order to reach the same contradiction. Thus, every periodic point of f\u0101\x88\x88int{\mathcal{OS}}_{\mathrm{\u012a\xbc}}(M) is hyperbolic.āā”
End of the proof of Theorem 1.2 Let f\u0101\x88\x88int{\mathcal{OS}}_{\mathrm{\u012a\xbc}}(M). By Proposition 2.3, we see that f\u0101\x88\x88{\mathcal{F}}_{\mathrm{\u012a\xbc}}(M). Thus, by Theorem 1.1, f is Anosov.āā”
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Acknowledgements
We wish to thank the referee for carefully reading of the manuscript and providing us many good suggestions. 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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Lee, K., Lee, M. Volume preserving diffeomorphisms with orbital shadowing. J Inequal Appl 2013, 18 (2013). https://doi.org/10.1186/1029242X201318
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DOI: https://doi.org/10.1186/1029242X201318