Generic Accessibility (part 1) – Andy Hammerlindl

Pugh-Shub Conjecture: Generic measure preserving partially hyperbolic diffeomorphism is ergodic. [PS (2000)]

The accessibility approach breaks this into two conjectures (and both are open):

Conjecture A: Generic partially hyperbolic diffeomorphism (measure preserving or not) is accessible.

Conjecture B: All measure preserving accessible partially hyperbolic diffeomorphisms are ergodic.

Note that if both conjecture A and conjecture B are true, then the Pugh-Shub Conjecture is true, but the failure of either won’t imply the conjecture being wrong.

Here we discuss the recent result of RHRHU (2008) which proves Pugh-Shub conjecture in the case where $\dim (E^c)=1$ by using accessibility.

We are going to focus on the proof of Conjecture A, here’s a sketch of proof of Conjecture B when $\dim (E^c)=1$ is assumed:

Theorem B: Let $M$ be compact manifold, $f: M \rightarrow M$ be a measure preserving accessible partially hyperbolic diffeomorphism, $\dim (E^c)=1$ , then $f$ is ergodic.

Proof: Let $\phi: M \rightarrow \mathbb{R}$ be $f$ invariant

Let $A = \phi^{-1}(( - \infty, c])$ , if $m(A)>0$ then $\exists p \in A$ s.t. $p$ is a density point of $A$ .

At this point there are some technical details in the paper which we are going to skip, but the main idea is to the fact that $\dim (E^c)=1$ (or in this case even the weaker hypothesis center brunching would work) to prove that in our case $y \in M$ is a density point iff $y$ is a “leaf density point” in both its center-stable and center-unstable leaves. Hence by accessibility from $p$ to $y$ , we can “push” the point $p$ along the us-path that joins $p$ to $y$ and induce that $y$ is a “leaf density point” in $A$ hence a density point in $A$ .

$\therefore$ all points $y \in M$ are density points of $A$ hence $m(A)=1$ .

Note that here if we replace accessibility by essential accessibility, we still get $m(A)=1$ .

Hence $\forall c \in \mathbb{R}$ , either $m(\phi^{-1}(( - \infty, c]))=0$ or $m(\phi^{-1}(( - \infty, c])) = 1$

$\therefore \ \phi$ is essentially constant. $\therefore f$ is ergodic.

This establishes theorem B.

Let $PH^r(M)$ be the set of measure preserving diffeomorphisms on $M$ that are of class $C^r$

Theorem A: Accessibility is open dense in the space of diffeomorphisms in $PH^r(M)$ with $\dim(E^c) = 1$ .

For any $x \in M$ , let $AC(x)$ denote the set of points that’s accessible from $x$

Let $\mathcal{D} = \{ f \in {PH}^r (M) | \forall \ x \in Per(f), AC(x)$ is open $\}$

Fact: $\mathcal{D} \subseteq PH^r(M)$ with $\dim(E^c) = 1$ is $G_\delta$ and $\mathcal{D} = \mathcal{A} \sqcup \mathcal{B}$
where $\mathcal{A} = \{ \ f \ | \ f$ is accessible $\}$ and
$\mathcal{B} = \{ \ f \ | \ per(f) = \phi$ and $E^u \oplus E^s$ is integrable $\}$

Note that this actually requires some rather technical work which was done in the paper, here we skip the proof of this.

Let $U(f) = \{ \ x \in M \ | \ AC(x)$ is open $\}$

It’s easy to see that $U(f)$ is automatically open hence $V(f) = M \setminus U(f)$ is compact.

Proposition: Let $x \in M$ , the following are equivalent:

1) $AC(x)$ has non-empty interior

2) $AC(x)$ is open

3) $AC(x) \cap \mathcal{W}^c_{loc}(x)$ has non-empty interior in $\mathcal{W}^c_{loc}(x)$

Proof: 1) $\Rightarrow$ 2) $\Rightarrow$ 3) $\Rightarrow$ 1)

Mainly by drawing pictures and standard topology.

Unweaving lemma: $\forall x \in Per(f), \ \exists g \in$ latex PH^r(M)$ with $\dim(E^c) = 1$ s.t. the $C^r$ distance between $f$ and $g$ is arbitrarily small, $x \in Per(g)$ and $AC_g(x)$ is open.

The proof is left to the second part of the talk…

2 thoughts on “Generic Accessibility (part 1) – Andy Hammerlindl”

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Generic Accessibility (part 1) – Andy Hammerlindl

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