Stably asymptotic average shadowing property and dominated splitting

Abstract

Let f be a diffeomorphism of a closed n-dimensional C^∞ manifold. In this article, we show that C¹-generically, if f has the C¹-stably asymptotic average shadowing property on a closed set then it admits a dominated splitting.

Mathematical Subject Classification: 34D05; 37C20; 37D30.

1 Introduction

The notion of the pseudo-orbits very often appears in several branches of the modern theory of dynamical system. For instance, the pseudo-orbit property (shadowing property) usually plays an important role in stability theory In this article, we consider the asymptotic average shadowing property, which was introduced in Gu [1], is a special version of the shadowing property We find a relation between the stably asymptotic average shadowing property (on manifold) and the dominated splitting structure on the vector bundle. In differentiable dynamical system, dominated splitting on the vector bundle is a nature generalization of hyperbolicity and is investigated by many mathematicians [2–11].

Here we denote M a closed n-dimensional smooth manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C¹-topology. Denote by d the distance on M induced from a Riemannian metric ||⋅|| on the tangent bundle TM. Let f ∈ Diff(M). A sequence ${\left\{x_i\right\}}_{i=-\infty }^{\infty }$ in M is called an asymptotic average pseudo orbit of f if

$$\underset{n\to \infty }{\text{lim}}\frac{1}{2n}\sum _{i=-n}^{n-1}d\left(f\left(x_i\right),x_{i+1}\right)=0.$$

An asymptotic average pseudo orbit {x _i }_i∈ℤis said to be asymptotically shadowed in average by the point z if

$$\underset{n\to \infty }{\text{lim}}\frac{1}{2n}{\displaystyle \sum _{i=-n}^{n-1}d(f^i(z),(x_i)=0.}$$

Given an invariant set Λ of f, we say f has the asymptotic average shadowing property on Λ if for any asymptotic pseudo orbit {x _i }_i∈ℤ, there exist a point z ∈ Λ which asymptotically shadows {x _i }_i∈ℤ.

Let f ∈ Diff(M), and let Λ be a closed f-invariant set. We say that Λ is locally maximal if there is a compact neighborhood U of Λ such that ∩ _{n ∈ℕ} fⁿ (U) = Λ. Now we can introduce a notion of C¹-stably the asymptotic average shadowing property on a locally maximal invariant set.

Definition 1.1 Let Λ be a locally maximal invariant set of f ∈ Diff(M). We say that f has the C^l-stably asymptotic average shadowing property on Λ, ( or Λ is C^l-stably asymptotic average shadowable with respect to f ) if there are a compact neighborhood U of f and a C^l-neighborhood $\mathcal{U}\left(f\right)$ of f such that Λ = Λ _f (U) = ∩ _{n ∈ℤ} fⁿ(U) (locally maximal), and for any $g\in \mathcal{U}\left(f\right),g{|}_{{\Lambda }_g\left(U\right)}$ has the asymptotic average shadowing property, where Λ _g (U) = ∩ _{n ∈ℤ} gⁿ(U) is the continuation of Λ.

Let Λ ⊂ M be an f-invariant closed set. We say that Λ admits a dominated splitting if the tangent bundle T_ΛM has a continuous Df-invariant splitting E ⊕ F and there exist constants C > 0 and 0 < λ < 1 such that

$$\Vert D_x{f}^n|E_{(x)}\Vert .\Vert D_x{f}^{-n}|{}_{F(f^n(x))}\Vert \le C{\lambda }^n$$

for all x ∈ Λ and n ≥ 0.

The following remark gives an equivalent definition of dominated splitting.

Remark 1.2 Let Λ be a closed f-invariant set. A splitting T_ΛM = E ⊕ F is called a l-dominated splitting for a positive integer l if E and F are Df-invariant and

$$\Vert D{f}^l|{}_{E(x)}\Vert /m(D{f}^l|{}_{F(x)})\le \frac{1}{2},$$

for all x ∈ Λ, where m(A) = inf{||Aυ||: ||υ|| = 1} denotes the minimum norm of a linear map A.

Now we can state main results of this article.

Theorem 1.3 Let Λ be a closed set of f ∈ Diff(M). Then C¹-generically, if f has the C¹-stably asymptotic average shadowing property on Λ then it admits a dominated splitting.

Theorem 1.4 Let Λ be a transitive set. If f has the C¹-stably asymptotic average shadowing property on Λ then it admits a dominated splitting.

2 Proof of theorems

Theorems 1.3 and 1.4 are all base on the following proposition:

Proposition 2.1 Let Λ be a closed locally manximal invariant set of f, if f has the C¹-stably asymptotic average shadowing property on Λ, and there exist a sequence g _n goes to f and periodic orbits P _n of g _n which converges to Λ in Hausdorff limits, then Λ admits a dominated splitting.

Firstly, we give the notation of pre-sink (resp. pre-source) which prevent the stably asymptotic average shadowing property. A periodic point p of f is called a pre-sink (resp. pre-source) if Df^π(p)(p) has a multiplicity one eigenvalue with modulus 1 and the other eigenvalues has norm strictly less than 1 (resp. bigger than 1).

Lemma 2.2 Let Λ be a closed set of f. Suppose that f_|Λ has the C¹-stably asymptotic average shadowing property. Let U and $\mathcal{U}\left(f\right)$ be given in the Definition 1.1, then for any $g\in \mathcal{U}\left(f\right)$, g has neither pre-sink nor pre-sources with the orbit staying in U.

Proof. We prove the lemma by contradiction. Assume that there is $g\in \mathcal{U}\left(f\right)$ such that g has a pre-sink p with Orb(p) ⊂ U.

By the Franks' Lemma, we can linearize g at p with respect to the exponential coordinates exp _p , i.e., after an arbitrarily small perturbation, we can get a diffeomorphism $g_1\in \mathcal{U}\left(f\right)$ such that there is ϵ₁ > 0 small enough with $B_{{\epsilon }_1}\left(\text{Orb}\left(p\right)\right)\subset U$ such that

$$g_1{\left|{}_{B_{{\in }_1}(g^i(p))}={\exp }_{g^{i+1}(p)}\circ D_{g^i(p)}g\circ {\exp }_{g^i(p)}^{-1}\right|}_{B_{{\in }_1}(g^i(p))},$$

for any 0 ≤ i ≤ π(p) - 1.

Since p is pre-sink of g, D _p g^π(p)has a multiplicity one eigenvalue such that | λ| = 1 and other eigenvalues of D _p g^π(p)have moduli less than 1. Denote by $E_p^c$the eigenspace corresponding to λ, and $E_p^s$the eigenspace corresponding to the eigenvalues with modulus less than 1. Thus $T_{p}M=E_p^c\oplus E_p^s$. If λ ∈ ℝ then dim$E_p^c=1$, and if λ ∈ ℂ then dim$E_p^c=2$.

At first, we consider the case dim $E_p^c=1$. For simplicity, we suppose that λ = 1, and $g_1^{\pi \left(p\right)}\left(p\right)=p$. The case of λ = -1 can be proved similarly. Since the eigenvalue λ = 1, there is a small arc ${ℐ}_p\subset B_{{\epsilon }_1}\left(p\right)\cap {\text{exp}}_p\left(E_p^c\left({\epsilon }_1\right)\right)$centered at p such that $g_1^{\pi \left(p\right)}{|}_{{ℐ}_p}$is the identity map. Here $E_p^c\left({\epsilon }_1\right)$is the ϵ₁-ball in $E_p^c$center at the origin O _p .

There exist D > 0 such that for any z ∈ B _D (p), there exists $x\in {ℐ}_p$ such that $g_1^{n\pi \left(p\right)}\left(z\right)\to x$as n → ∞. Take two distinct points $a,b\in {ℐ}_p$ such that d(a, b) = D/4.

We construct an asymptotic average pseudo orbit of g₁ as follows.

$$\begin{array}{c}{x}_{-i}=g_1^{-i}a,\\ x_0=a,x_1=g_1\left(a\right),\dots ,x_{\pi \left(p\right)-1}=g_1^{\pi \left(p\right)-1}a,x_{\pi \left(p\right)}=b,\dots ,\\ x_{\left(2^k-2\right)\pi \left(p\right)}=a,x_{\left(2^k-2\right)\pi \left(p\right)+1}=g_1\left(a\right),\dots ,x_{\left(2^k+2^{k-1}-2\right)\pi \left(p\right)-1}=g_1^{-1}a,\\ x_{\left(2^k+2^{k-1}-2\right)\pi \left(p\right)}=b,\dots ,x_{\left(2^{k+1}-2\right)\pi \left(p\right)-1}=g_1^{-1}\left(b\right),\dots .\end{array}$$

One can easily check that ξ = {x _i }_i∈ℤis an asymptotic average pseudo orbit of g₁.

Since g₁ has the asymptotic average shadowing property on ${\Lambda }_{g_1}\left(U\right)$, we can find a point z such that the point z is shadows ξ = {x _i }_i∈ℤin asymptotic average, i.e.,

$$\underset{n\to \infty }{\text{lim}}\frac{1}{2n}\sum _{i=-n}^{n-1}d\left(g_1^i\left(z\right),x_i\right)=0.$$

It is easy to see that there is n₀ > 0 such that $g_1^{n_0}\left(z\right)\in B_D\left(p\right)$. Hence there exists a point $x\in {ℐ}_p$ such that $g_1^{n\pi \left(p\right)+n_0}\left(z\right)\to x$, as n → ∞. From the choice of a, b and the fact that $g_1^{\pi \left(p\right)}{|}_{{ℐ}_p}=Id$, we have

$$\underset{n\to \infty }{\text{lim}}\frac{1}{n}\sum _{i-0}^{n-1}d\left(g_1^i\left(z\right),x_i\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{n}\sum _{i=0}^{n-1}d\left(g_1^{i-n0}\left(x\right),x_i\right)>0.$$

This is a contradiction.

Finally, we consider the case dim $E_p^c=2$. There is a disk ${\mathcal{D}}_p\subset B_{{\epsilon }_1}\left(p\right)\cap {\text{exp}}_p\left(E_p^c\left({\epsilon }_1\right)\right)$centered at p such that $g_1^{\pi \left(p\right)}{|}_{{\mathcal{D}}_p}$is a rotation. Note that $\mathcal{D}$ consists of $g_1^{\pi \left(p\right)}$-invariant circles. We take a and b in different circles. Then by similar arguments as above, we get the contradiction. We omit the details and finish the proof here.

Let GL(n) be the group of linear isomorphisms of ℝⁿ. A sequence ξ : ℤ → GL(n) is called periodic if there is k > 0 such that ξ _{j +k}= ξ _j for k ∈ ℤ. We call a finite subset $\mathcal{A}=\left\{{\xi }_i:0\le i\le k-1\right\}\subset GL\left(n\right)$ is a periodic family with period k. For a periodic family $\mathcal{A}=\left\{{\xi }_i:0\le i\le n-1\right\}$, we denote ${\mathcal{C}}_{\mathcal{A}}={\xi }_{n-1}\circ {\xi }_{n-2}\circ \cdots \circ {\xi }_0$.

Definition 2.3 We say that the periodic family $\mathcal{A}=\left\{{\xi }_i:0\le i\le n-1\right\}$ admits an l-dominated splitting, if there is a splitting ℝⁿ= E ⊕ F which satisfies:

(a) E and F are ${\mathcal{C}}_{\mathcal{A}}$ invariant, i.e., ${\mathcal{C}}_{\mathcal{A}}\left(E\right)=E$ and ${\mathcal{C}}_{\mathcal{A}}\left(F\right)=F$,

(b) For any k = 0,1,2,...,

$$\frac{∥{\xi }_{k+l-1}\circ \cdots \circ {\xi }_{k+1}\circ {\xi }_k{|}_{E_k}∥}{m\left({\xi }_{k+l-1}\circ \cdots \circ {\xi }_{k1}\circ {\xi }_k{|}_{F_k}\right)}\le \frac{1}{2},$$

where E _k = ξ _{k- 1} ○ ξ _{k- 2} ○ ⋯ ○ ξ₀(E) and F _k = ξ _{k- 1} ○ ξ _{k- 2} ○ ⋯ ○ ξ₀(F).

We know the following theorems for periodic family from [4] which is useful for our result.

Theorem 2.4 Given any ϵ > 0 and K > 0, there is positive integers n₂ ≥ 0 and l ≥ 0 which satisfies the following property: given any periodic family $\mathcal{A}=\left\{{\xi }_i:0\le i\le n-1\right\}$ which satisfies the period n ≥ n₂ and max $\left\{∥{\xi }_i∥,∥{\xi }_i^{-1}∥\right\}\le K$, for all i = 0,1,...,n-1, if $\mathcal{A}$ does not admits any l-dominated splitting, then one can find a periodic family $ℬ=\left\{{\zeta }_0,{\zeta }_1,...,{\zeta }_{n-1}\right\}$ such that $\text{max}\left\{∥{\zeta }_i-{\xi }_i∥,∥{\zeta }_i^{-1}-{\xi }_i^{-1}∥\right\}<\epsilon$for any i = 0,1,...,n-1, and $\text{det}\left({\mathcal{C}}_{\mathcal{A}}\right)=\text{det}\left({\mathcal{C}}_{ℬ}\right)$ and the eigenvalues of ${\mathcal{C}}_{ℬ}$are all real, and have same modulus.

To prove Theorem 2.4, we need another lemma about uniformly contracting family. Let $\mathcal{A}=\left\{{\xi }_i:0\le i\le k-1\right\}\subset GL\left(n\right)$ be a periodic family. We say the sequence $\mathcal{A}$ is uniformly contracting family if there is a constant δ > 0 such that for any δ-perturbation of $\mathcal{A}$ are sink, i.e., for any $ℬ=\left\{{\xi }_i:0\le i\le k-1\right\}$ with ||ζ _i - ζ _i || < δ, all eigenvalue of ${\mathcal{C}}_{ℬ}$ have moduli less than 1. Similarly, we can define the uniformly expanding periodic family. The following theorem is well known.

Theorem 2.5 [12] For any δ > 0 and K > 0, there are constants C > 0,0 < λ < 1 and positive integer m such that if $\mathcal{A}=\left\{A_0,A_1,...,A_{n-1}\right\}$ is a uniformly contracting periodic family which satisfies

$$\text{max}\left\{∥A_i∥,∥A_i^{-1}∥\right\}<K$$

for any i = 0,1,...,n - 1 and n > m, then

$$\prod _{j=0}^{k-1}∥\prod _{i=0}^{m-1}{A}_{i+mj}∥\le C{\lambda }^k,$$

where k = [n/m].

Now we return to our main proposition, the Proposition 2.1. Let P _n be given as in

Proposition 2.1. Choose p _n ∈ P _n , then we get a linear map sequence

$${\mathcal{A}}_n=\left\{D_{p_n}f,D_{f\left(p_n\right)}f,\dots ,D_{f^{\pi \left(p_n\right)-1}\left(p_n\right)}f\right\}.$$

Lemma 2.6 [[10], Lemma 3.2.] If Λ is not a periodic orbit and ${\mathcal{A}}_n$ is given in above. Then for any ϵ > 0 there exists an n₀(ϵ) > 0 such that for any n > n₀(ϵ), ${\mathcal{A}}_n$ is neither ϵ-uniformly contracting nor ϵ-uniformly expanding.

Since the proof is essentially the same as that of [10], we omit the proof here. From the above lemma and main conclusion of [4], one can get the following lemma. The proof of the following can be found in [10].

Lemma 2.7 [[10], Lemma 3.3.] Let Λ, g _n and P _n be given as in the assumption of Proposition 2.1. Then for any ϵ > 0 there are n(ϵ),l(ϵ) > 0 such that for any n > n(ϵ) if P _n does not admit an l(ϵ) dominated splitting, then one can find $g_n^{\prime }$C¹ϵ-close g _n and preserving the orbit of P _n such that P _n is pre-sink or pre-source respecting $g_n^{\prime }$.

From the above lemmas and the next property of dominated splitting, we can get Proposition 2.1.

Lemma 2.8 [[3], Lemma 1.4.] Let g _n converges to f and if Λ _n be a closed g _n -invariant set such that the Hausdorff limit of Λ _n equal to Λ. If ${\Lambda }_{g_n}\left(U\right)$ admits a l-dominated splitting respecting g _n , then Λ admits an l-dominated splitting respecting f.

Now we can get our Theorems 1.3 and 1.4, Theorem 1.3 follows two results:

Lemma 2.9 [1, 13] Let Λ be a closed set of f ∈ Diff(M). If f has the asymptotic average shadowing property on Λ then Λ is a chain transitive set.

The following Lemma is in [14].

Lemma 2.10 There is a residual set $\mathcal{G}\subset \text{Di}\left(M\right)$ such that for any $f\in \mathcal{G}$, a compact f-invariant set Λ is a chain transitive set if and if Λ is a sequence {P _n } of periodic orbits of f with the Hausdorff topology.

Theorem 1.4 follows the result:

Lemma 2.11 [[11], Corollary 2.7.1.] Let Λ be a transitive set. Then there are a sequence {g _n } of diffeomorphism and a sequence {P _n } of periodic orbits of g _n with period π(P _n ) → ∞ such that g _n → f in the C^l-topology and P _n → _H Λ as n → ∞, where → _H is the Hausdorff limit, and π(P _n ) is the period of P _n .