On today’s seminar I presented that fact that an accessible pair of 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 , and be a continuous map with . Then .

Proof. Let . We need to show that there exists such that . To this end we consider following map . Then for each we have

So .

By Brouwer Fixed Point Theorem we see there exists with . In particular we have . This finishes the proof.

So if is close to identitical map on , its image still contains an open ball: it can not be nowhere dense.

2. Every accessible class is a -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 and consider the sequence of sets where

.

.

where for odd and for even.

Similarly we can define by revising the roles of s and u of . Note that . So it suffices to consider the sequences of . Clearly we have

.

We will inductively show that is compact.

It is true for . Assume is compact, and let . We will show there exists an open nbhd with .

By definition of , (otherwise for some in that intersected set.)

Both and being compact subsets, we can pick an open nbhd with . By continuity of the (stable or unstable) invariant foliations,

and the continuous dependence of their induced Riemannian metrics on the leaves, for each in a small nbhd of . In particular .

Hence the set on the left hand side is disjoint from . This proves that is also closed.