Accessibility

On today’s seminar I presented that fact that an accessible pair of $C^1$ distributions is stable accessible (Theorem 3.4 of Pugh, Shub: Stably Ergodic Dynamical Systems and Partial Hyperbolicity). There are several notes.

1. Continuous maps close to identity.

GAN Shaobo showed that one step among the proof is true for general case.

Let $\epsilon\in(0,1)$, $D^n=\{x\in\mathbb{R}^n:|x|\le1\}$ and $f:\mathbb{D}^n\to\mathbb{R}^n$ be a continuous map with $d(fx,x)\le\epsilon$. Then $f(D^n)\supset B(0,1-\epsilon)$.

Proof. Let $p\in B(0,1-\epsilon)$. We need to show that there exists $y\in D^n$ such that $fy=p$. To this end we consider following map $g:\mathbb{D}^n\to\mathbb{R}^n,y\mapsto p+y-fy$. Then for each $y\in D^n$ we have

$|gy|\le |p|+|fy-y|\le|p|+\epsilon<1.$ So $g(D^n)\subset \text{cl}(B(0,|p|+\epsilon))\subset D^n$.

By Brouwer Fixed Point Theorem we see there exists $y\in D^n$ with $y=gy=p+y-fy$. In particular we have $fy=p$. This finishes the proof.

So if $f$ is $C^0$ close to identitical map on $D^n$, its image still contains an open ball: it can not be nowhere dense.

2. Every accessible class $A_f(x)$ is a $F_\sigma$-set (hence measurable).
(the proof is from Lemma 8.4 of Avila, Santamaria. and Viana: Cocycles over partially hyperbolic maps)
To this end we first fix $L\in\mathbb{N}$ and consider the sequence of sets $K_{n,L}$ where
$K_{1,L}=W^s(x,L)=\{y\in W^s(x): d_{s}(x,y)\le L\}$.
$K_{n,L}=\{y\in M:y\in W^{\sigma(n)}(z,L)\text{ for some }z\in K_{n-1,L}\}$.
where $\sigma(n)=s$ for $n$ odd and $\sigma(n)=u$ for $n$ even.

Similarly we can define $T_{n,L}$ by revising the roles of s and u of $\sigma$. Note that $T_{n,L}\subset K_{n+1,L}$. So it suffices to consider the sequences of $K$. Clearly we have
$A_f(x)=\bigcup_{n,L\in\mathbb{N}}K_{n,L}$.

We will inductively show that $K_{n,L}$ is compact.
It is true for $n=1$. Assume $K_{n-1,L}$ is compact, and let $z\notin K_{n,L}$. We will show there exists an open nbhd $U\ni x$ with $U\cap K_{n,L}=\emptyset$.
By definition of $K_{n,L}$, $W^{\sigma(n)}(z,L)\cap K_{n-1,L}=\emptyset$ (otherwise $z\in W^{\sigma(n)}(p,L)\subset K_{n,L}$ for some $p$ in that intersected set.)
Both $W^{\sigma(n)}(z,L)$ and $K_{n-1,L}$ being compact subsets, we can pick an open nbhd $U\supset W^{\sigma(n)}(z,L)$ with $U\cap K_{n,L} =\emptyset$. By continuity of the (stable or unstable) invariant foliations,
and the continuous dependence of their induced Riemannian metrics on the leaves, $W^{\sigma(n)}(w,L)\subset U$ for each $w\in V$ in a small nbhd $V$ of $z$. In particular $U\cap K_{n,L}=\emptyset$.
Hence the set on the left hand side is disjoint from $K_{n-1,L}$. This proves that $K_{n,L}$ is also closed.