The ergodic shadowing property from the robust and generic view point

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R E S E A R C H

Open Access

In this paper, we discuss that if a diﬀeomorphisms has theC1_{-stably ergodic}

shadowing property in a closed set, then it is a hyperbolic elementary set. Moreover,

C1_{-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

Keywords: ergodic shadowing; shadowing; locally maximal; generic; Anosov

1 Introduction

LetMbe a closedC∞ manifold, and letDiff(M) be the space of diﬀeomorphisms ofM endowed with theC_{-topology. Denote by}_d_{the distance on}_M_{induced from a}

Rieman-nian metric · on the tangent bundleTM. Let f ∈Diff(M). Forδ> , a sequence of ing property. The shadowing property usually plays an important role in the investiga-tion of stability theory and ergodic theory. For instance, Sakai [] proved that iff has the C_{-robustly shadowing property, then}_f _{is structurally stable. Now we introduce the}

no-tion of the ergodic shadowing property which was introduced and studied by []. Lee has shown in [] that iff belongs to theC_{-interior of the set of all diﬀeomorphisms}

having the ergodic shadowing property, then it is structurally stable diﬀeomorphisms. In [], Lee showed that iff is local star condition and has the ergodic shadowing property on the homoclinic class, then it is hyperbolic. For anyδ> , a sequenceξ ={xi}i∈Zis a

Here#Ais the number of elements of the setA. We say thatf has theergodic shadowing propertyin(orf| hasergodic shadowing) if for any> , there is aδ>  such that

everyδ-ergodic pseudo orbitξ ={xi}i∈Z⊂off there is a pointz∈such that, for

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Ns+

Note thatf has the ergodic shadowing property onandf has the ergodic shadowing property inare diﬀerent notions. That is, the shadowing point is inMor. In the ﬁrst notion, the shadowing point is inM. In the second notion, the shadowing point is in. In this paper we consider the latter case.

We say thatislocally maximalif there is a compact neighborhoodUofsuch that

n∈Z

fn(U) =f(U) =.

Now, we introduce the notion of theC_{-stably ergodic shadowing property in a closed set.}

Deﬁnition . Letbe a closedf-invariant set. We say thatf has theC_{-stably ergodic}

shadowing propertyinif

We say thatishyperbolicif the tangent bundleTMhas aDf-invariant splittingEs⊕

Eu_{and there exist constants}_C_{>  and  <}_λ_{<  such that} elementary set) iff| is transitive (resp. mixing) and locally maximal. Note that ifis

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 .[, Theorem .] Letbe a closed f -invariant set.If f has the C-stably ergodic shadowing property in,then it is a hyperbolic elementary set.

Corollary . If f belongs to the C_{-interior of the set of all diﬀeomorphisms having the}

ergodic shadowing property,then it is transitive Anosov.

We say that a subsetG⊂Diff(M) isresidualifGcontains the intersection of a countable family of open and dense subsets ofDiff(M); in this caseGis dense inDiff(M). A property P is said to beC_-generic_{if P holds for all diﬀeomorphisms which belong to some residual}

subset ofDiff(M). We use the terminology ‘forC_-generic_f_{’ to express ‘there is a residual}

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Recently, Ahnet al.[] have given a partial answer which isC_{-generically: if a locally}

maximal homoclinic class is shadowing, then it is hyperbolic. Lee has shown in [] that C_{-generically: if}_f _{has the limit shadowing property on the homoclinic class, then it is}

hyperbolic. Inspired by this, we consider thatC-generically:fhas the ergodic shadowing property in a locally maximal closed set. Then we have the following.

Theorem . For C-generic f,if f has the ergodic shadowing property in a locally maxi-mal closed set,then it is a hyperbolic elementary set.Moreover,C_{-generically:}_{if f has}

the ergodic shadowing property,then it is transitive Anosov.

2 Proof of Theorem 1.4

LetP(f) be the set of periodic points off. Iff|is transitive, then everyp∈∩P(f) is

saddle, that is, there is no eigenvalues ofDpfπ(p)with modulus equal to , at least one of them is greater than , at least one of them is smaller than , whereπ(p) is the minimum period ofp.

Lemma .[, Corollary .] If f has the ergodic shadowing property in,then f| is

mixing.

By Lemma .,f has the ergodic shadowing property in, thenf|is mixing, and so

f|is transitive. Thusp∈∩P(f) is neither a sink nor a source.

Lemma .[, Lemma .] If f has the ergodic shadowing property in,then f has a ﬁnite shadowing property in.

We say thatf has theﬁnite shadowing propertyonif for any>  there isδ>  such that, for any ﬁniteδ-pseudo orbit{x,x, . . . ,xn} ⊂, there isy∈Msuch thatd(fi(y),xi) <

for all ≤i<n. In [, Lemma ..], Pilyugin showed thatf has a ﬁnite shadowing shad-owing property on, thenf has the shadowing property on.

Lemma . Let f have the ergodic shadowing inandbe locally maximal in U.Then the shadowing point taken from.

Proof Letf have the ergodic shadowing property in, and letU be a locally maximal of. For any> , letδ>  be the number of the ergodic shadowing property off. Take a sequenceγ ={xi}n_i_=(n≥) such thatγ is aδ-pseudo orbit off andγ ⊂. As in the proof of [, Lemma .], there is aδ-pseudo orbitη={xi}

i=nsuch thatη⊂. Then we set

ξ ={. . . ,γ,η,γ,η, . . .}is aδ-ergodic pseudo orbit off. Clear thatξ ⊂. Sincef has the ergodic shadowing property in,ξ can be ergodic shadowed by some pointy∈. By Lemma ., there isγ ∈ξ such thatd(fi_(y),_{xi) <}_{for }_≤_i_≤_n_{– . By [, Lemma ..],}_f has the shadowing property on. Sinceis locally maximal inU, the shadowing point

y∈.

Letp∈P(f) be a hyperbolic saddle with periodπ(p) > . Then there are the local stable manifoldWs

(p) and the local unstable manifoldWu(p) ofpfor some=(p) > . It is

easily seen that ifd(fn_(x),_fn_(p))_≤_{for all}_n_≥_{, then}_x_∈_Ws

(p), and ifd(fn(x),fn(p))≤

for alln≤, thenx∈Wu

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deﬁned as following. It is well known that if pis a hyperbolic periodic point of f with periodk, then the sets

Ws(p) =x∈M:fkn(x)→pasn→ ∞ and

Wu(p) =x∈M:f–kn(x)→pasn→ ∞

areC-injectively immersed submanifolds ofM.

Lemma . Let p,q∈P(f)be hyperbolic saddles.If f has the ergodic shadowing property in a closed set,then Ws_(p)_∩_Wu_(q)₌_∅_,_{and W}u_(p)_∩_Ws_(q)₌_∅_.

Proof Letp,q∈P(f) be hyperbolic saddles, and letU be a locally maximal neighbor-hood of. Suppose thatf has the ergodic shadowing property in a locally maximal. Since p and q are hyperbolic, there are (p) >  and (q) >  as in the above. Take

=min{(p),(q)}/ and let  <δ≤ be the number of the ergodic shadowing property off. For simplicity, we may assume thatf(p) =pandf(q) =q. Sincefhas the ergodic shad-owing property in,f|is chain transitive. Then we can construct a ﬁniteδ-pseudo orbit

formptoqas follows:x=p,xn=q(n≥), andd(f(xi),xi+) <δfor all  <i<n– . Put

(i)x–i=f–i(p), for alli≤, and (ii)xn+i=fi(q) for alli≥. Then we have the sequence

ξ ={xi}i∈Z={. . . ,p,x,x, . . . ,xn,xn+, . . .}. It is clearly aδ-ergodic pseudo orbit off. Since

f has the ergodic shadowing property inand locally maximal, by Lemma .,f has the ﬁnite shadowing property onand so, by [, Lemma ..],f has the shadowing property in. By the shadowing property in, we can show thatOrb(y)⊂Wu_(p)_∩_Ws_{(q), and so}

Wu(p)∩Ws(q)=∅. The other case is similar.

A diﬀeomorphismf isKupka-Smaleif their periodic points off are hyperbolic and if p,q∈P(f), thenWs_{(p) is transversal to}_Wu_{(q). Then it is}_C_{-residual in}_Diff_{(M). Denote}

byKS(M) the set of all Kupka-Smale diﬀeomorphisms. The following was proved by [].

Lemma .[, Lemma .] Letbe locally maximal in U,and letU(f)be given.If for any g∈U(f),p∈g(U)∩P(g)is not hyperbolic,then there is g∈U(f)such that g has

two hyperbolic periodic points p,q∈g(U)with diﬀerent indices.

Denote byF(M) the set off ∈Diff(M) such that there is aC_neighborhood_U_(f_{) of}

f such that, for anyg∈U(f), everyp∈P(g) is hyperbolic. In [], Hayashi proved that f ∈F(M) if and only iff satisfies both Axiom A and the no-cycle condition. We say that f isthe local star condition diffeomorphismif there exist aC-neighborhoodU(f) and a neighborhoodUofsuch that, for anyg∈U(f), everyp∈g(U)∩P(g) is hyperbolic (see []). Denote byF() the set of all local star diffeomorphisms. Note that there are aC_{-neighborhood}_U_(f_{) and a neighborhood}_U_of_p_{such that, for all}_g_∈_U_(f_{), there is a}

unique hyperbolic periodic pointpg∈Uofg with the same period aspandindex(pg) =

index(p). Hereindex(p) =dimEs

p, and the pointpgis called thecontinuationofp.

Lemma .[, Lemma .] There is a residual setG⊂Diff(M)such that,for any f∈G,

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Lemma . There is a residual set G⊂Diff(M)such that,for any f ∈G,if f has the

ergodic shadowing property in a locally maximal,then for any p,q∈∩P(f)

index(p) =index(q).

Proof Letf ∈G=G∩KS(M), and letp,q∈∩P(f) be hyperbolic saddles. Suppose

thatf has the ergodic shadowing property in a locally maximal. Then by Lemma . Ws(p)∩Wu(q)=∅andWu(p)∩Ws(q)=∅. Sincef∈G,Ws(p)Wu(q)=∅andWu(p)

Ws(q)=∅. This means thatp∼qand soindex(p) =index(q).

Letpbe a periodic point off. For  <δ< , we say thatphas aδ-weak eigenvalueif Dfπ(p)_{(p) has an eigenvalue}_λ_{such that ( –}_δ₎π(p)_<_|_λ_|_{< ( +}_δ₎π(p)_{. We say that a periodic}

point has areal spectrumif all of its eigenvalues are real and simple spectrumif all its eigenvalues have multiplicity one. Denote byPh(f) the set of all hyperbolic periodic points off.

Lemma .[, Lemma .] There is a residual setG⊂Diff(M)such that,for any f∈G:

• For anyδ> ,if for anyC_{-neighborhood}_U_(f₎_of_f _{there exist}_g_∈_U_(f₎_and_pg_∈_Ph_(g)

with aδ-weak eigenvalue,then there isp∈Ph(f)with aδ-weak eigenvalue.

• For anyδ> ,ifq∈Ph(f)with aδ-weak eigenvalue and a real spectrum,then there is p∈Ph(f)with aδ-weak eigenvalue with a simple real spectrum.

Lemma . There is a residual setG⊂Diff(M)such that,for any f ∈G,if f has the

ergodic shadowing property in a locally maximal,then there existsη> such that,for any q∈∩Ph(f),q has noη-weak eigenvalues.

Proof Letf ∈G=G∩Ghave the ergodic shadowing property in a locally maximal.

We will derive a contradiction. Suppose that, for anyη> , there isq∈∩Ph(f) such that qhas anη-weak eigenvalue. By Franks’ lemma, there isg C_{-close to}_f _{such that}_p_{is not}

hyperbolic. By Franks’ lemma and Lemma ., there ish C_-nearby_g _and_C_{-close to}_f

such thathhas tow hyperbolic periodic pointsqh,γhwith diﬀerent indices. Sincef ∈G,

and it is locally maximal, by Lemma .f has two hyperbolic periodic pointsq,γ in. Sincef has the ergodic shadowing property in, this is a contradiction by Lemma ..

Proposition . There is a residual setG⊂Diff(M)such that,for any f ∈G,if f has

the ergodic shadowing property in,then f ∈F().

Proof Letf ∈Ghave the ergodic shadowing property in a locally maximal. Suppose by

contradiction thatf ∈/F(). Then there areg C_{-close to}_f _and_q_∈_{P(g) such that}_q_{has a} η-weak eigenvalue. Then by Lemma ., we get a contradiction. Thusf ∈F().

Proposition .[, Proposition A] Let f ∈G,and letbe locally maximal.If f has

the ergodic shadowing property in,then there are m> ,C≥andλ∈(, )such that, for any p∈∩P(f)withπ(q) >m,we have

π(p)–

i=

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π(p)–

i=

Dfm|_Es₍_fmi₍_x₎₎≤Cλπ(p) and Dfm|Es₍_x₎·Df–m|_Eu₍_fm₍_x₎₎≤λ,

whereπ(p)is the period of p.

Remark . By Pugh’s closing lemma, there is a residual setG⊂Diff(M) such that, for

anyf ∈G, iff|is transitive, then there is a periodic orbitpnsuch thatOrb(pn)→in

Hausdorﬀ metric.

Lemma .[, Theorem .] There is residual setG⊂Diff(M)such that,for any f ∈G,

for any ergodic measureμof f,there is a sequence of the periodic point pnsuch thatμpn→μ in weak∗topology andOrb(pn)→Supp(μ)in Hausdorﬀ metric.

The following was proved by Mañé []. Denote byM(f|) the set of invariant

proba-bilities on the Borelσ-algebra ofendowed with the weak∗topology.

Lemma . Let⊂M be a closed f -invariant set of f and E⊂TM be a continuous

invariant subbundle.If there is m> such that

logDfm|Edμ< 

for every ergodicμ∈M(fm_|

),then E is contracting.

Proof of Theorem. Letf ∈G∩G∩Ghave the ergodic shadowing property in a

lo-cally maximal. Then by Proposition ., we know thatadmits a dominated split-tingTM=E⊕F. Sincef has the ergodic shadowing property in, by Lemma .,

Re-mark ., and Lemma ., there is a sequence of periodic points such thatOrb(pn)→

Supp(μ) =in the Hausdorﬀ metric. By Proposition ., we have Dfm|Edμ= lim

n→∞ Df

m_|

Edμpn< .

By Lemma .,Eis contracting. Similarly, we can show thatFis expanding.

Corollary . For C_{-generic f}_,_{if f has the ergodic shadowing property,}_{then f is transitive}

Anosov.

Competing interests

The author declares that they have no competing interests.

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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