Tag Archives: accessible

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.

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