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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