Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers

Abstract

For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton—a set consisting of finitely many hyperbolic periodic points with maximal cardinality for which there exist no heteroclinic intersections. We build the one-to-one corresponding between periodic points in any skeleton and physical measures. By making perturbations on skeletons, we study the continuity of physical measures with respect to dynamics under $C^1$-topology.

Appendix A: Proof of Proposition 3.5

Thanks to the Pliss-Like Lemma founded by Andersson–Vasquez [8, Lemma A], which implies the following result(see [18, Lemma 5.8]).

Lemma A.1

Given constants $C_1<C_2\le \max \{0,C_2\}<C$, for any $\varepsilon >0$, there is $\rho =\rho (C_1,C_2,C,\varepsilon )>0$ such that for any sequence $\{a_n\}_{n\in \mathbb {N}}\subset \mathbb {R}$ satisfying:

• $|a_n|\le C$, $\forall n\in \mathbb {N}$,

• there is a subset $P\subset \mathbb {N}$ such that $\mathcal {D}_{L}(P)>1-\rho$ and $a_n\le C_1$ for any $n\in P$,

then there is a subset $Q \subset \mathbb {N}$ with $\mathcal {D}_{U}(Q)>1-\varepsilon$ such that for any $j\in Q$, one has that

$$\begin{aligned} \sum _{i=0}^{n-1}a_{i+j}\le nC_2,~~~\forall n\in \mathbb {N}. \end{aligned}$$

The following lemma plays a key role in the proof of Proposition 3.5.

Lemma A.2

Let f be a $C^1$ diffeomorphism with dominated splitting $TM=E\oplus F$. Given $0<\alpha <\alpha _0$, if $\mu$ is an f-invariant measure such that

$$\begin{aligned}&\lim _{n\rightarrow +\infty }\frac{1}{n}\log \Vert Df^{-n}|_{E(x)}\Vert <-\alpha _0, \quad \mu -a.e. ~x\in M; \end{aligned}$$

(A.1)

$$\begin{aligned}&\lim _{n\rightarrow +\infty }\frac{1}{n}\log \Vert Df^{n}|_{F(x)}\Vert <-\alpha _0, \quad \mu -a.e. ~x\in M. \end{aligned}$$

(A.2)

Then, for every $\varepsilon >0$ there exists $\ell \in \mathbb {N}$ and a neighborhood $\mathcal {U}\times \mathcal {V}$ of $(f,\mu )$ such that for every $g\in \mathcal {U}$ and g-invariant measure $\nu \in \mathcal {V}$, we have

$$\begin{aligned} \nu \left( \Lambda _{\ell }(g,\alpha ,E,F)\right) >1-\varepsilon . \end{aligned}$$

Proof

For simplicity, for every g that is $C^1$-close to f, we introduce the two blocks $\Lambda _{\ell }^{-}(g,\alpha )$ and $\Lambda _{\ell }^{+}(g, \alpha )$ defined as follows:

$$\begin{aligned} \Lambda _{\ell }^{-}(g, \alpha )= & {} \left\{ x: \frac{1}{n\ell }\sum _{i=0}^{n-1}\log \Vert Dg^{-\ell }|_{E(g^{-i\ell }(x))}\Vert \le -\alpha , \quad \forall n\in \mathbb {N}\right\} ;\\ \Lambda _{\ell }^{+}(g,\alpha )= & {} \left\{ x: \frac{1}{n\ell }\sum _{i=0}^{n-1}\log \Vert Dg^{\ell }|_{F(g^{i\ell }(x))}\Vert \le -\alpha ,\quad \forall n\in \mathbb {N}\right\} . \end{aligned}$$

Hence, $\Lambda _{\ell }(g,\alpha )=\Lambda _{\ell }^{-}(g, \alpha )\cap \Lambda _{\ell }^{+}(g,\alpha )$. Given $n\in \mathbb {N}$, let us define

$$\begin{aligned} \Lambda ^E_{g,n}= & {} \left\{ x: \frac{1}{n}\log \Vert Dg^{-n}|_{E(x)}\Vert<-\alpha _0 \right\} ;\\ \Lambda ^F_{g,n}= & {} \left\{ x: \frac{1}{n}\log \Vert Dg^{n}|_{F(x)}\Vert <-\alpha _0 \right\} . \end{aligned}$$

Take $\Lambda _{g,n}=\Lambda ^E_{g,n}\cap \Lambda ^F_{g,n}$.

Choose $\varepsilon '>0$ satisfying $(1-\varepsilon ')^2>1-\varepsilon /2$ and fix constants

$$\begin{aligned} C_1=-\alpha _0, \quad C_2=-\alpha , \quad C=\max _{x\in M}|\log \Vert Df^{\pm 1}|_{E(x)}\Vert |+1. \end{aligned}$$

Then take $\rho =\rho (C_1,C_2,C,\varepsilon )$ as in Lemma A.1. It follows from (A.1) and (A.2) that for $0<\rho \varepsilon '<1$, there exists $\ell \in \mathbb {N}$ such that $\mu (\Lambda _{f,\ell })>1-\rho \varepsilon '$. By the continuity of diffeomorphisms and [8, Lemma 3.2], there exists an open neighborhood $\mathcal {U}\times \mathcal {V}$ of $(f,\mu )$ such that

$$\begin{aligned} \nu (\Lambda _{g,\ell })>1-\rho \varepsilon '. \end{aligned}$$

(A.3)

Moreover, we can assume $\max _{x\in M}|\log \Vert Dg^{\pm 1}|_{E(x)}\Vert |\le C$ for every $g\in \mathcal {U}$.

Take any $g\in \mathcal {U}$ and g-invariant measure $\nu$ in $\mathcal {V}$. Hence, we have $\nu (\Lambda ^E_{g,\ell })>1-\rho \varepsilon '$, $\nu (\Lambda ^F_{g,\ell })>1-\rho \varepsilon '$.

Since $\nu$ is $g^{\ell }$-invariant, by Birkhoff’s ergodic theorem we get that for $\nu$-almost every $x\in M$, there exists the limit

$$\begin{aligned} \phi (x)=\lim _{n\rightarrow +\infty }\frac{1}{n}\# \left\{ i: 0 \le i \le n-1, g^{-\ell i}(x) \in \Lambda ^E_{g,\ell } \right\} . \end{aligned}$$

Furthermore, applying (A.3) we have

$$\begin{aligned} \int \phi (x)d\nu = \nu (\Lambda ^E_{g,\ell })>1-\rho \varepsilon '. \end{aligned}$$

(A.4)

Let $G=\left\{ x: 1-\rho <\phi (x)\le 1\right\}$, we claim that $\nu (G)>1-\varepsilon '$. Indeed, observe that $M\setminus G = \{x: \rho \le 1-\phi (x) \le 1\},$ this together with (A.4) imply

$$\begin{aligned} \rho \varepsilon '> \int (1-\phi )d\nu \ge \int (1-\phi )\chi _ {M\setminus G}d\nu \ge \rho \nu (M\setminus G). \end{aligned}$$

Thus, $\nu (G)>1-\varepsilon '$. For any $x\in G$, we define

$$\begin{aligned} a_i=\frac{1}{\ell }\log \Vert Dg^{-\ell }|_{E(g^{-i\ell }x)}\Vert ,\quad \forall i\ge 0, \end{aligned}$$

and

$$\begin{aligned} P=\{i\ge 0: g^{-i\ell }(x)\in \Lambda ^E_{g,\ell }\}. \end{aligned}$$

Then, by properties of $\Lambda ^E_{g,\ell }$ and G, one has that $a_i<-\alpha$ for every $i\in P$, and $\mathcal {D}_{L}(P)>1-\rho$. Thus, by applying Lemma A.1, there exists a subset $Q\subset \mathbb {N}$ with density estimate

$$\begin{aligned} \mathcal {D}_{U}(Q)>1-\varepsilon '. \end{aligned}$$

(A.5)

Moreover, for every $j\in Q$, we have

$$\begin{aligned} \sum _{i=0}^{n-1}a_{i+j}\le -n\alpha , \quad \forall n\in \mathbb {N}, \end{aligned}$$

which can be rewrite as

$$\begin{aligned} \prod _{i=0}^{n-1}\Vert Dg^{-\ell }|_{E(g^{-{i+j}\ell }(x))}\Vert \le \mathrm{e}^{-n\ell \alpha }, \quad \forall n\in \mathbb {N}. \end{aligned}$$

Therefore, by combining (A.5) with Birkhoff’s ergodic theorem we know that for $\nu$-almost every $x\in G$,

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{1}{n}\#\left\{ i\in [0,n-1]: g^{-i\ell }(x)\in \Lambda _{\ell }^{-}(g,\alpha )\right\} >1-\varepsilon '. \end{aligned}$$

Therefore, there exists $n_0\in \mathbb {N}$ and a subset

$$\begin{aligned} G_{0}=\left\{ x\in G: \frac{1}{n_0}\#\left\{ i\in [0, n_0-1]: g^{-i\ell }(x)\in \Lambda _{\ell }^{-}(g,\alpha )\right\} >1-\varepsilon ' \right\} \end{aligned}$$

satisfying $\nu (G_0)>1-\varepsilon '$. Then, we have

$$\begin{aligned} \nu (\Lambda _{\ell }^{-}(g,\alpha ))= & {} \int \frac{1}{n_0}\sum _{i=0}^{n_0-1}\chi _{\Lambda _{\ell }^{-}(g,\alpha )}(g^{-i\ell }x)d\nu \\\ge & {} \int _{G_0} \frac{1}{n_0}\sum _{i=0}^{n_0-1}\chi _{\Lambda _{\ell }^{-}(g,\alpha )}(g^{-i\ell }x)d\nu \\= & {} \int _{G_0} \frac{1}{n_0}\#\left\{ i\in [0, n_0-1]: g^{-i\ell }(x)\in \Lambda _{\ell }^{-}(g,\alpha )\right\} d\nu \\\ge & {} (1-\varepsilon ')\nu (G_0)\\> & {} (1-\varepsilon ')^2>1-\varepsilon /2. \end{aligned}$$

Similarly, for F direction one can get $\nu (\Lambda _{\ell }^{-}(g,\alpha ))>1-\varepsilon /2$. Therefore,

$$\begin{aligned} \nu (\Lambda _{\ell }(g,\alpha ))>1-\varepsilon . \end{aligned}$$

The proof of Lemma A.2 is complete. $\square$

By Lemma 2.8, one can conclude the following result. It can be derived from [43, Lemma 3.4] and [43, Lemma 3.5].

Lemma A.3

If $f\in PH^1_{EC}(M)$, then there exist $N\in \mathbb {N}$ and $\beta >0$ such that for every $\mu \in G^u(f)$, one has

$$\begin{aligned} \lim _{n\rightarrow +\infty } \frac{1}{n}\sum _{i=0}^{n-1}\log \Vert Df^{-N}|_{E^{cu}(f^{-iN}(x))}\Vert <-\beta \end{aligned}$$

for $\mu$-almost every x.

Now we can finish the proof of Proposition 3.5.

Proof of Proposition 3.5

By Lemma A.3, there exist $N\in \mathbb {N}$ and $\beta >0$ such that for every $\mu \in G^u(f)$ and $\mu$-almost every x, we have

$$\begin{aligned} \lim _{n\rightarrow +\infty } \frac{1}{n}\sum _{i=0}^{n-1}\log \Vert Df^{-N}|_{E^{cu}(f^{-iN}(x))}\Vert <-\beta . \end{aligned}$$

By taking $\alpha _{0}=\beta /N$, we get that for $\mu$-almost every x, it has that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{1}{n}\log \Vert Df^{-n}|_{E^{cu}(x)}\Vert <-\alpha _0. \end{aligned}$$

Following the same argument, up to reducing $\alpha _0$, for any Gibbs u-state $\mu$ and $\mu$-almost every x, we have

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{1}{n}\log \Vert Df^{n}|_{E^{cs}(x)}\Vert <-\alpha _{0}. \end{aligned}$$

Now fix $\alpha \in (0,\alpha _0)$. By Lemma A.2, for every $\varepsilon >0$, for any $\mu \in G^u(f)$, there exists $\ell _{\mu }\in \mathbb {N}$ and neighborhood $\mathcal {U}_{\mu }\times \mathcal {V}_{\mu }$ of $(f,\mu )$ in $G^u(PH^{1}_{EC}(M))$⁸ such that

$$\begin{aligned} \nu (\Lambda _{\ell _{\mu }}(g, \alpha ))>1-\varepsilon \end{aligned}$$

(A.6)

for every $(g,\nu )\in \mathcal {U}_{\mu }\times \mathcal {V}_{\mu }$. By compactness of $G^u(f)$ as stated in Item (1) of Lemma 2.8, there exist finitely many Gibbs u-states $\mu _1,\ldots ,\mu _j$ such that the corresponding neighborhoods $\mathcal {V}_i{:}{=}\mathcal {V}_{\mu _i}, 1\le i \le j$ form a covering of $G^u(f)$. Writing $\mathcal {U}_i{:}{=}\mathcal {U}_{\mu _i}$ and $\ell _i{:}{=}\ell _{\mu _i}$ for every $1\le i \le j$. Let

$$\begin{aligned} \mathcal {U}=\bigcap _{1\le i \le j} \mathcal {U}_i; \quad \mathcal {V}=\bigcup _{1\le i\le j}\mathcal {V}_i;\quad \ell =\prod _{i=1}^j \ell _j. \end{aligned}$$

Note that by definition of $\ell$, we have $\Lambda _{\ell _i}(g,\alpha )\subset \Lambda _{\ell }(g,\alpha )$ for every $1\le i \le j$. By Lemma 2.9, up to shrinking $\mathcal {U}$ we assume $G^u(g)\subset \mathcal {V}$ for every $g\in \mathcal {U}$. Therefore, for every $(g,\nu )\in \mathcal {U}\times \mathcal {V}$, there is some $1\le i\le j$ such that $(g,\nu )\in \mathcal {U}_i\times \mathcal {V}_i$, and thus (A.6) implies $\nu (\Lambda _{\ell }(g,\alpha ))>1-\varepsilon$. $\square$