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Chain components with stably limit shadowing property are hyperbolic
Advances in Difference Equations volume 2014, Article number: 104 (2014)
Abstract
Let f be a diffeomorphism on a closed smooth manifold M. In this paper, we show that f has the {C}^{1}stably limit shadowing property on the chain component {C}_{f}(p) of f containing a hyperbolic periodic point p, if and only if {C}_{f}(p) is a hyperbolic basic set.
MSC:37C50, 34D10.
1 Introduction
Various closed invariant sets (transitive set, chain transitive set, homoclinic class, chain component, etc.) in dynamical systems are natural candidates to replace Smale’s hyperbolic basic sets in nonhyperbolic theory of differentiable dynamical systems see [1–6]). To investigate the above, we deal with the shadowing property. It usually plays an important role in the stability theory and ergodic theory (see [7]).
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). Let 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. 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. We write x\leftrightsquigarrow y if x\u21ddy and y\u21ddx. The set of points \{x\in M:x\leftrightsquigarrow x\} is called the chain recurrent set of f and is denoted by \mathcal{R}(f). Denote {C}_{f}(p)=\{x\in M:x\u21ddp\text{and}p\u21ddx\} the chain component of f containing p. For a closed finvariant set \mathrm{\Lambda}\subset M, we say that Λ is chain transitive if for any point x,y\in \mathrm{\Lambda} and \delta >0, there exists a δpseudo orbit {\{{x}_{i}\}}_{i={a}_{\delta}}^{{b}_{\delta}}\subset \mathrm{\Lambda} ({a}_{\delta}<{b}_{\delta}) of f such that {x}_{{a}_{\delta}}=x and {x}_{{b}_{\delta}}=y.
Let \mathrm{\Lambda}\subset M be a closed finvariant set. 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 b.
Now, we introduce the limit shadowing property which was introduced and studied by Lee [8]. We say that f has the limit shadowing property on Λ if there exists \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}. Here, the sequence {\{{x}_{i}\}}_{i\in \mathbb{Z}} is called a δlimit pseudo orbit of f. It is easy to see that f has the limit shadowing property on Λ if and only if {f}^{n} has the limit shadowing property on Λ for n\in \mathbb{Z}\setminus \{0\}, and the identity map does not have the limit shadowing property.
Note that the above definition is not the shadowing property, also it is not the notion of the original limit shadowing property in (see [[8], Examples 3, 4] and [7, 9]).
We say that Λ is locally maximal if there is a compact neighborhood U of Λ such that {\bigcap}_{n\in \mathbb{Z}}{f}^{n}(U)=\mathrm{\Lambda}. We say that f has the {C}^{1}stably limit shadowing property on Λ if there are a {C}^{1}neighborhood \mathcal{U}(f) of f and a compact neighborhood U of Λ such that

(1)
\mathrm{\Lambda}={\mathrm{\Lambda}}_{f}(U)={\bigcap}_{n\in \mathbb{Z}}{f}^{n}(U) (locally maximal),

(2)
for any g\in \mathcal{U}(f), g has the limit shadowing property on {\mathrm{\Lambda}}_{g}(U), where {\mathrm{\Lambda}}_{g}(U)={\bigcap}_{n\in \mathbb{Z}}{g}^{n}(U) is the continuation of \mathrm{\Lambda}={\mathrm{\Lambda}}_{f}(U).
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. A point x\in {W}^{s}(p)\cap {W}^{u}(p) is called a homoclinic point of f associated to p, and it is said to be a transversal homoclinic point of f if the above intersection is transverse. The closure of the homoclinic points of f associated to p is called the nontransversal homoclinic class of f associated to p, say, generalized homoclinic class, and it is denoted by {\overline{H}}_{f}(p), and the closure of the transversal homoclinic points of f associated to p is called the transversal homoclinic class of f associated to p, and it is denoted by {H}_{f}(p). Let p, q be hyperbolic periodic points of f. We say that p and q are homoclinically related, and write p\sim q if
It is clear that if p\sim q then index(p)=index(q); i.e., dim{W}^{s}(p)=dim{W}^{s}(q). By Smale’s transverse homoclinic point theorem, {H}_{f}(p) coincides with the closure of the set of hyperbolic periodic points q of f such that p\sim q. In this paper, we consider all periodic points of the saddle type, because, if p\in P(f) is a sink or a source, then {C}_{f}(p) is the periodic orbit of p itself.
Note that if p is a hyperbolic periodic point of f then there is a neighborhood U of p and a {C}^{1}neighborhood \mathcal{U}(f) of f such that for any g\in \mathcal{U}(f), there exists a unique hyperbolic periodic point {p}_{g} of g in U with the same period as p and index({p}_{g})=index(p). Such a point {p}_{g} is called the continuation of p={p}_{f}.
Let Λ be a closed finvariant set. 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. Moreover, we say that Λ admits a dominated splitting if the tangent bundle {T}_{\mathrm{\Lambda}}M has a continuous Dfinvariant splitting E\oplus F and there exist constants C>0 and 0<\lambda <1 such that
for all x\in \mathrm{\Lambda} and n\ge 0.
The following is the main theorem in this paper.
Theorem 1.1 Let p be a hyperbolic periodic point of f, and let {C}_{f}(p) be the chain component of f associated to p. Then f has the {C}^{1}stably limit shadowing property on {C}_{f}(p) if and only if {C}_{f}(p) is hyperbolic.
Let Λ be a locally maximal subset of M. In [8], Lee showed that if Λ is hyperbolic then it is limit shadowable. Note that a hyperbolic set Λ has the local product structure if and only if it is locally maximal. Since the chain component {C}_{f}(p) has the local product structure, if {C}_{f}(p) is hyperbolic, {C}_{f}(p) is locally maximal. Thus by the hyperbolicity of the chain component {C}_{f}(p), f has the {C}^{1}stably limit shadowing property. Thus, in this paper, we show that if f has the {C}^{1}stably limit shadowing property on {C}_{f}(p), then {C}_{f}(p) is hyperbolic.
2 Proof of Theorem 1.1
Let M be as before, and let f\in Diff(M).
Lemma 2.1 Let Λ be a locally maximal subset of M. If f has the limit shadowing property on Λ then the shadowing points are taken from Λ.
Proof Let \delta >0 be the number of the limit shadowing property of f, and let U be a locally maximal neighborhood of Λ. Suppose that f has the limit shadowing property on Λ. Let {\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda} be a δlimit pseudo orbit of f. To derive a contradiction, we may assume that there is y\in M\setminus \mathrm{\Lambda} such that
Since Λ is compact, there is \eta >0 such that {B}_{\eta}(\mathrm{\Lambda})\subset U, where {B}_{\eta}(\mathrm{\Lambda}) is a ηneighborhood of Λ. Since {\{{x}_{i}\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda} and by the limit shadowing property, we can find l\in \mathbb{Z} such that {f}^{l}(y)\in {B}_{\eta}(\mathrm{\Lambda}). Since Λ is locally maximal in U and finvariant,
Then for all n\in \mathbb{Z}, {f}^{n}({f}^{l}(y))={f}^{l+n}(y)\in \mathrm{\Lambda}. Since Λ is finvariant, y\in {f}^{nl}(\mathrm{\Lambda})=\mathrm{\Lambda}, this is a contradiction. Thus the limit shadowing points are in Λ. □
Let us recall some notions for the proof of the following lemma. A compact invariant set Λ is attracting if \mathrm{\Lambda}={\bigcap}_{n\ge 0}{f}^{n}(U) for some neighborhood U of Λ satisfying {f}^{n}(U)\subset U for all n>0. An attractor of f is a transitive attracting set of f and a repeller is an attractor for {f}^{n}. We say that Λ is a proper attractor or repeller if \mathrm{\varnothing}\ne \mathrm{\Lambda}\ne M. A sink (source) of f is an attracting (repelling) critical orbit of f.
Lemma 2.2 ([[10], Proposition 3])
Let Λ be a locally maximal set. f{}_{\mathrm{\Lambda}} is chain transitive if and only if Λ has no proper attractor for f.
Lemma 2.3 Let Λ be a locally maximal set. If f has the limit shadowing property on Λ then f{}_{\mathrm{\Lambda}} is chain transitive.
Proof Suppose Λ has a proper attractor P in Λ. Then P\ne \mathrm{\varnothing} and \mathrm{\Lambda}\setminus P\ne \mathrm{\varnothing}. Since P is an attractor, there exists \delta >0 such that P attracts the open \delta /2neighborhood {B}_{\delta /2}(P) of P in Λ. Choose q\in \mathrm{\Lambda}\setminus {B}_{\delta /2}(P) and p\in P such that d(q,p)<\delta. Consider a sequence
with i\in \mathbb{Z}. Clearly, the sequence {\{{x}_{i}\}}_{i\in \mathbb{Z}} is a δlimit pseudo orbit of f in Λ. Then by Lemma 2.1, there is y\in \mathrm{\Lambda} such that
Then there exists N>0 large enough such that {f}^{N}(y)\in {B}_{\delta /2}(P). Therefore, {f}^{n}({f}^{N}(y))\in {B}_{\delta /2}(P) for n>0, since P is an attractor. Taking N={i}_{k}, we have that y={f}^{{i}_{k}}({f}^{{i}_{k}}(y))\in {B}_{\delta /2}(P). Thus, by definition of {B}_{\delta /2}(P), we have that
This contradicts the definition of the limit shadowing property and completes the proof. □
Lemma 2.4 Let Λ be a locally maximal set. Suppose f has the limit shadowing property on Λ. Then for any hyperbolic periodic points p, q in Λ,
Proof Suppose f has the limit shadowing property on locally maximal Λ, and let p,q\in \mathrm{\Lambda} be hyperbolic periodic points for f. We will show that {W}^{s}(p)\cap {W}^{u}(q)\ne \mathrm{\varnothing}. Other case is similar. Since f has the limit shadowing property on locally maximal Λ, by Lemma 2.3, we can take a δchain {\{{x}_{i}\}}_{i=0}^{n} from p to q such that {x}_{0}=p, {x}_{n}=q. Then we can construct a δlimit pseudo orbit ξ as follows: (i) {x}_{i}={f}^{i}(p), i<0, (ii) d(f({x}_{i}),{x}_{i+1})<\delta, i=0,\dots ,n1 and (iii) {x}_{n+i}={f}^{i}(q), i\ge 0. Then
Clearly, ξ is a δlimit pseudo orbit of f in Λ. Then, by Lemma 2.1, there exists a point y\in \mathrm{\Lambda} such that
This implies that y\in {W}^{u}(p) and {f}^{n}(y)\in {W}^{s}(q) (y\in {W}^{s}(q)). Thus {W}^{u}(p)\cap {W}^{s}(q)\ne \mathrm{\varnothing}. □
The following socalled Franks lemma will play essential roles in our proof.
Lemma 2.5 Let \mathcal{U}(f) be any given {C}^{1}neighborhood of f. Then there exist \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.
Proof See the proof of Lemma 1.1 [11]. □
Lemma 2.6 ([[12], Lemma 2.4])
Let Λ be locally maximal in U, and let \mathcal{U}(f) be given. If p\in {\mathrm{\Lambda}}_{g}(U)\cap P(g) (g\in \mathcal{U}(f)) is not hyperbolic, then there is {g}_{1}\in \mathcal{U}(f) possessing hyperbolic periodic points {q}_{1} and {q}_{2} in {\mathrm{\Lambda}}_{{g}_{1}}(U) with different indices.
In this section, we will prove Theorem 1.1 by making use of the technique developed by Mañé in [13]. That is, we use the notion of uniform hyperbolicity for a family of periodic sequences of linear isomorphisms of {\mathbb{R}}^{dimM}. For this, we need several lemmas.
We say that a diffeomorphism f is KupkaSmale if for any periodic point of f is hyperbolic and their invariant manifolds intersect transversely and denote the set of KupkaSmale diffeomorphisms by \mathcal{KS}(M). It is well known that \mathcal{KS}(M) is residual in Diff(M).
Lemma 2.7 Let f\in Diff(M), and let Λ be a closed finvariant set. Suppose that f has the {C}^{1}stably limit shadowing property on Λ. Then there exist a {C}^{1}neighborhood \mathcal{U}(f) of f and a compact 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 for g, where {\mathrm{\Lambda}}_{g}(U)={\bigcap}_{n\in \mathbb{Z}}{g}^{n}(U).
Proof Since f has the {C}^{1}stably limit shadowing property on Λ, there exist a {C}^{1}neighborhood \mathcal{U}(f) of f and a compact neighborhood U of Λ such that for any g\in \mathcal{U}(f), g has the limit shadowing property on {\mathrm{\Lambda}}_{g}(U)={\bigcap}_{n\in \mathbb{Z}}{g}^{n}(U). Let \u03f5>0 and {\mathcal{U}}_{0}(f)\subset \mathcal{U}(f) be the corresponding number and {C}^{1}neighborhood of f given by Lemma 2.5 with respect to \mathcal{U}(f). Suppose there is a point q\in {\mathrm{\Lambda}}_{g}(U)\cap P(g) which is not hyperbolic. Then by Lemma 2.6, we can choose {g}_{1}\in {\mathcal{U}}_{0}(f) such that index{p}_{{g}_{1}}\ne index{q}_{{g}_{1}}, where {p}_{{g}_{1}},{q}_{{g}_{1}}\in {\mathrm{\Lambda}}_{{g}_{1}}(U)\cap P({g}_{1}). Then dim{W}^{s}({p}_{{g}_{1}})+dim{W}^{u}({q}_{{g}_{1}})<dimM or dim{W}^{u}({p}_{{g}_{1}})+dim{W}^{s}({q}_{{g}_{1}})<dimM. We may assume that dim{W}^{s}({p}_{{g}_{1}})+dim{W}^{u}({q}_{{g}_{1}})<dimM. By Lemma 2.5, we can take h\in \mathcal{U}({g}_{1})\cap \mathcal{KS}(M) such that index({p}_{{g}_{1}})=index({p}_{h}) and index({q}_{{g}_{1}})=index({q}_{h}) where {p}_{h}, {q}_{h} are the continuation of {p}_{{g}_{1}}, {q}_{{g}_{1}} for h, respectively. Then, since h is KupkaSmale, {W}^{s}({p}_{h})\cap {W}^{u}({q}_{h})=\mathrm{\varnothing}. On the other hand, since h\in \mathcal{U}(f), h{}_{{\mathrm{\Lambda}}_{h}(U)} satisfies the limit shadowing property so that {W}^{s}({p}_{h})\cap {W}^{u}({q}_{h})\ne \mathrm{\varnothing} by Lemma 2.4. This is a contradiction and completes the proof. □
It is a wellknown result that the transversal homoclinic class {H}_{f}(p) is a subset of the generalized homoclinic class {\overline{H}}_{f}(p), and it is a subset of the chain component {C}_{f}(p). However, under the notion of the limit shadowing property with locally maximal, {\overline{H}}_{f}(p)={C}_{f}(p). It is obtained by the following lemma.
Lemma 2.8 Let U be a locally maximal neighborhood of {C}_{f}(p). If f has the limit shadowing property on {C}_{f}(p) then {C}_{f}(p)={\overline{H}}_{f}(p).
Proof Let p be a hyperbolic saddle. For simplify we may assume that f(p)=p. Let U be a locally maximal neighborhood of {C}_{f}(p). Suppose that f has the limit shadowing property on a locally maximal {C}_{f}(p). For any x\in {C}_{f}(p), we show that x\in {\overline{H}}_{f}(p). Let \delta >0 be the number of the limit shadowing property of f. Since x\leftrightsquigarrow p, there is a periodic δpseudo orbit {\{{x}_{i}\}}_{i=l}^{k} of f such that {x}_{l}=p, {x}_{0}=x and {x}_{k}=p for some l=l(\delta ), k=k(\delta )>0. Then the periodic δpseudo orbit {\{{x}_{i}\}}_{l}^{k}\subset {C}_{f}(p) (see [[14], Proposition 1.6]). Now we construct a δlimit pseudo orbit as follows: (i) {x}_{li}={f}^{i}(p) for all i\ge 0, and (ii) {x}_{k+i}={f}^{i}(p) for all i\ge 0. Then we know the δlimit pseudo orbit
Since {C}_{f}(p) is locally maximal, by Lemma 2.1, for small \eta >0 we can take a point y\in {C}_{f}(p) such that d(x,y)<\eta and d({f}^{i}(y),{x}_{i})\to 0 as i\to \pm \mathrm{\infty}. Since d({f}^{i}(y),{x}_{i})\to 0 as i\to \pm \mathrm{\infty}, we know
Furthermore, by Theorem 7.3 in [15], we see that y\in {B}_{\eta}(x) where {B}_{\eta}(x) denotes the ηneighborhood of x. Thus we conclude that
This means {C}_{f}(p)\subset {\overline{H}}_{f}(p), and therefore {C}_{f}(p)={\overline{H}}_{f}(p). □
It is well known that a dominated splitting is always extended to a neighborhood. More precisely, let Λ be a closed finvariant set. Then if Λ admits a dominated splitting {T}_{\mathrm{\Lambda}}M=E\oplus F such that dim{E}_{x} (x\in \mathrm{\Lambda}) is constant, then there are a {C}^{1}neighborhood \mathcal{U}(f) of f and a compact neighborhood U of Λ such that for any g\in \mathcal{U}(f), {\bigcap}_{n\in \mathbb{Z}}{g}^{n}(U) admits a dominated splitting
with dim{E}^{\prime}(g)=dimE.
From Lemma 2.7, the family of periodic sequences of linear isomorphisms of {\mathbb{R}}^{dimM} generated by Dg (g\in {\mathcal{U}}_{0}(f)) along the hyperbolic periodic points p\in {\mathrm{\Lambda}}_{g}(U)\cap P(g) is uniformly hyperbolic. That is, there exists \u03f5>0 such that for any g\in {\mathcal{U}}_{0}(f), p\in {\mathrm{\Lambda}}_{g}(U)\cap P(g), and any sequence of linear maps {L}_{i}:{T}_{{g}^{i}(p)}M\to {T}_{{g}^{i+1}(p)}M with \parallel {L}_{i}{D}_{{g}^{i}(p)}g\parallel <\u03f5 for 0\le i\le \pi (p)1, {\prod}_{i=0}^{\pi (p)1}{L}_{i} is hyperbolic. Here {\mathcal{U}}_{0}(f) is the {C}^{1}neighborhood of f given by Lemma 2.7. Thus by Proposition II.1 in [13] and Lemma 2.7 above, we get the following proposition.
Proposition 2.9 Suppose that f has the {C}^{1}stably limit shadowing property on the chain component {C}_{f}(p) of f associated to a hyperbolic periodic point p and let {\mathcal{U}}_{0}(f) as Lemma 2.7. Then there are constants C>0, \lambda \in (0,1) and m>0 such that

(a)
for any g\in {\mathcal{U}}_{0}(f), if q\in {\mathrm{\Lambda}}_{g}(U)\cap P(g) has the minimum period \pi (q)\ge m, then
\prod _{i=0}^{k1}\parallel {D}_{{g}^{im}(q)}{g}^{m}{}_{{E}_{{g}^{im}(q)}^{s}}\parallel <C{\lambda}^{k}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\prod _{i=0}^{k1}\parallel {D}_{{g}^{im}(q)}{g}^{m}{}_{{E}_{{g}^{im}(q)}^{u}}\parallel <C{\lambda}^{k},where k=[\pi (q)/m], and {\mathrm{\Lambda}}_{g}(U)={\bigcap}_{n\in \mathbb{Z}}{g}^{n}(U).

(b)
{C}_{f}(p) admits a dominated splitting {T}_{{C}_{f}(p)}M=E\oplus F with dimE=index(p).
Remark From Proposition 2.9(b) and Lemma 2.8, {C}_{f}(p)={\overline{H}}_{f}(p)={H}_{f}(p).
In general, a nonhyperbolic homoclinic class {H}_{f}(p) contains saddle periodic points with different indices. Let p be a hyperbolic periodic point of f.
Proposition 2.10 Suppose that f has the {C}^{1}stably limit shadowing property on {C}_{f}(p). Then for any q\in {C}_{f}(p)\cap P(f),
where index(p)=dim{W}^{s}(p).
Proof Suppose that f has the {C}^{1}stably limit shadowing property on {C}_{f}(p). Let U be a compact neighborhood of {C}_{f}(p), and let \mathcal{U}(f) be a {C}^{1}neighborhood of f. Then for any g\in \mathcal{U}(f), g has the limit shadowing property on {\mathrm{\Lambda}}_{g}(U)={\bigcap}_{n\in \mathbb{Z}}{g}^{n}(U). By Lemma 2.7, for any q\in {C}_{f}(p)\cap P(f), q is hyperbolic. By contradiction, suppose that there is q\in {C}_{f}(p)\cap P(f) such that index(p)\ne index(q). This implies that
Then we can choose g\in \mathcal{U}(f)\cap \mathcal{KS}(M) such that index({p}_{g})=index(p) and index({q}_{g})=index(q) for the continuations {p}_{g},{q}_{g}\in {\mathrm{\Lambda}}_{g}(U)\cap P(g) of p, q, respectively. Then we may assume that dim{W}^{s}({q}_{g})+dim{W}^{u}({p}_{g})<dimM. Other case is similar. Since g is KupkaSmale, dim{W}^{s}({q}_{g})+dim{W}^{u}({p}_{g})<dimM implies that {W}^{s}({q}_{g})\cap {W}^{u}({p}_{g})=\mathrm{\varnothing}. On the other hand, by the definition of the {C}^{1}stably limit shadowing property, for {p}_{g},{q}_{g}\in {\mathrm{\Lambda}}_{g}(U)\cap P(g),
This is a contradiction and completes the proof. □
Note that for any hyperbolic periodic point q in {C}_{f}(p) for a hyperbolic periodic point p, there exist a {C}^{1}neighborhood \mathcal{U}(f) of f and a neighborhood U of {C}_{f}(p) such that for any g\in \mathcal{U}(f), there is unique {p}_{g}\in {C}_{g}({p}_{g})\cap P(g) which contained in {\mathrm{\Lambda}}_{g}(U)\cap P(g), where {p}_{g} is the continuation of p for g.
We denote the index(p) by j (0<j<dimM) and let {P}_{j}(f{}_{{H}_{f}(p)}) be the set of periodic points q\in {H}_{f}(p)\cap P(f) such that index(q)=j for all 0<j<dimM. Set {\mathrm{\Lambda}}_{j}(f)=\overline{{P}_{j}(f{}_{{H}_{f}(p)})}, then {H}_{f}(p)={\mathrm{\Lambda}}_{j}(f)={C}_{f}(p).
Lemma 2.11 Let {\mathcal{U}}_{0}(f) be the {C}^{1}neighborhood of f given by Lemma 2.7 and Proposition 2.9 and let \mathcal{V}(f)\subset {\mathcal{U}}_{0}(f) be a small connected {C}^{1}neighborhood of f. If g\in \mathcal{V}(f) satisfying g=f on M\setminus {U}_{j}, then
for any q\in {\mathrm{\Lambda}}_{g}(U)\cap {P}_{g}.
Proof Suppose the property is not true then there are {g}^{\prime}\in \mathcal{V}(f) and q\in {\mathrm{\Lambda}}_{{g}^{\prime}}\cap P({g}^{\prime}) such that {g}^{\prime}=f on M\setminus {U}_{j} and index(q)\ne index(p). Suppose that {({g}^{\prime})}^{n}(q)=q, {i}_{0}=index(q), and define \phi :\mathcal{V}(f)\to \mathbb{Z} by
where ♯A is the number of elements of A. By Lemma 2.7, the function φ is continuous, and since \mathcal{V}(f) is connected, it is constant. But the property of {g}^{\prime} implies \phi ({g}^{\prime})>\phi (f). This is a contradiction, so that the lemma is proved. □
For any \u03f5>0, denote by {B}_{\u03f5}(x,f) a ϵtubular neighborhood of forbit of x, that is,
We say that a point x\in M is well closable for f\in Diff(M) if for any \u03f5>0 there are g\in Diff(M) with {d}_{1}(f,g)<\u03f5 and p\in M such that p\in P(g), g=f on M\setminus {B}_{\u03f5}(x,f) and d({f}^{n}(x),{g}^{n}(p))\le \u03f5 for any 0\le n\le \pi (p), where \pi (p) is the period of p, and {d}_{1} is the {C}^{1}metric. Let {\mathrm{\Sigma}}_{f} denote the set of well closable points of f. Then we know the following fact.
Lemma 2.12 ([[13], Theorem A])
For any finvariant probability measure μ, we have \mu ({\mathrm{\Sigma}}_{f})=1.
Proof of Theorem 1.1 Suppose that f has the {C}^{1}stably limit shadowing property on {C}_{f}(p). Then there are a {C}^{1}neighborhood \mathcal{U}(f) of f and a compact neighborhood U of {C}_{f}(p) as in the definition. Let {\mathcal{U}}_{0}(f)\subset \mathcal{U}(f) of f given by Lemma 2.7 and Proposition 2.10. Define {\mathrm{\Lambda}}_{j} as the set such that every periodic orbit in it has index j. To get the conclusion, it is sufficient to show that {\mathrm{\Lambda}}_{j}(f) is hyperbolic since {H}_{f}(p)={C}_{f}(p)={\mathrm{\Lambda}}_{j}(f), where 0<j=index(p)<dimM. Now {C}_{f}(p) admits a dominated splitting {T}_{{C}_{f}(p)}M=E\oplus F such that dimE=index(p) by Proposition 2.9(b). Thus, as in the proof of [[13], Theorem B], we can show that
for all x\in {C}_{f}(p) and therefore the splitting is hyperbolic.
More precisely, we will prove the case of {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}\parallel {D}_{x}{f}^{n}{}_{E(x)}\parallel =0 (other case is similar). It is enough to show that for any x\in {C}_{f}(p), there exists n=n(x)>0 such that
If it is not true, then there is x\in {C}_{f}(p) such that
for all n\ge 0. Thus
for all n\ge 0.
From now, let {C}_{f}(p)=\mathrm{\Lambda}. Define a probability measure
Then there exists {\mu}_{{n}_{k}} (k\ge 0) such that {\mu}_{{n}_{k}}\to {\mu}_{0}\in {\mathcal{M}}_{{f}^{m}}(M), as k\to \mathrm{\infty}, where M is compact metric space. Thus
By Mañé ([13], p.521),
where {\mu}_{0} is a {f}^{m}invariant measure. Let
and {\mathrm{\Sigma}}_{f} as in Lemma 2.12.
Note that if x\notin P(f), 0\le \pi (y)=N such that d({f}^{N}(x),{f}^{N}(y))=d({f}^{N}(x),y)\to 0 as N\to \mathrm{\infty}, then d(x,y)\to 0. So it cannot be.
By Lemma 2.12, we know that for any \mu \in {\mathcal{M}}_{f}(M),
Then, for any \mu \in {\mathcal{M}}_{f}(\mathrm{\Lambda}),
since \mu ({C}_{f}(p))=1 and \mu ({\mathrm{\Sigma}}_{f})=1. Hence it defines an finvariant probability measure ν on {C}_{f}(p) by
Thus, {C}_{f}(p)={C}_{f}(p)\cap \mathrm{\Sigma}(f) almost everywhere. Therefore,
By Birkhoff’s theorem and the ergodic closing lemma, we can take {z}_{0}\in {C}_{f}(p)\cap \mathrm{\Sigma}(f) such that
By Proposition 2.9, this is a contradiction. Thus by Proposition 2.9, {z}_{0}\notin P(f).
Let C>0, m>0 and \lambda \in (0,1) be given by Proposition 2.9, and let us take \lambda <{\lambda}_{0}<1 and {n}_{0}>0 such that
Then, by Mañé’s ergodic closing lemma (Lemma 2.12), we can find g\in {\mathcal{V}}_{0}(f), g=f on M\setminus {U}_{j} and {z}_{g}\in {\mathrm{\Lambda}}_{g}(U)\cap P(g) nearby {z}_{0}. Moreover, we know that index({z}_{g})=index(p) since g=f on M\setminus {U}_{j}. By applying Lemma 2.5, we can construct {g}_{1}\in {\mathcal{V}}_{0}(f) (\subset \mathcal{V}(f)) {C}^{1}nearby g such that
(see [[13], pp.523524]). On the other hand, by Proposition 2.9, we see that
We can choose the period \pi ({z}_{{g}_{1}}) (>{n}_{0}) of {z}_{{g}_{1}} as large as {\lambda}_{0}^{k}\ge C{\lambda}^{k}. Here k=[\pi ({z}_{{g}_{1}})/m]. This is a contradiction. Thus,
for all x\in {C}_{f}(p). Therefore, {C}_{f}(p) is hyperbolic. This completes the proof of the ‘only if part’ of Theorem 1.1. □
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Acknowledgements
ML was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20110007649). JP was supported by BK21 math vision 2020 project.
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Lee, M., Park, J. Chain components with stably limit shadowing property are hyperbolic. Adv Differ Equ 2014, 104 (2014). https://doi.org/10.1186/168718472014104
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DOI: https://doi.org/10.1186/168718472014104
Keywords
 hyperbolic
 limit shadowing
 shadowing
 homoclinic class
 chain component