Tag: ergodic

Anosov flows

ConanLeave a comment

Amie told me today about their new result on perturbation of a volume-preserving Anosov flow in three dimensions. In order to not forget what it’s about, I decided to write a sketch of what I still remember here:

So, you are given a volume preserving Anosov flow in some three-manifold (and since it’s volume preserving and Anosov and three dimensional, of course we have one dimensional stable and unstable manifolds), let \varphi_1: M \rightarrow M be its time-1 map. Consider a C^\infty perturbation of \varphi_1. We are interested in when is the perturbed map still a time-1 map of a flow.

Note that we know partial hyperbolicity is an open property, our perturbed map will still be a partially hyperbolic diffeo. However in general it would no longer be a time 1 map of a flow. It turns out that we can tell whether or not it’s a time-1 map just by looking at the center foliation. (some condition to do with whether some measure on the center is atomic…I can’t recall)

Furthermore this infact don’t have much to do with the fact it’s a perturbation of the Anosov flow: we may start with any volume-preserving partially hyperbolic diffeomorphism in three-manifold M, assuming the diffeo preserves its center foliation (or more generally if it permutes each center leaf peroidically), then it’s time-one map of a flow precisely when their condition on the center foliation holds. Note that the center leaves are automatically preserved if the map was a perturbation of the Anosov flow.

Note that restriction our attention to volume preserving flows is essential in obtaining any of such results since in part it guarantees a dense set of periodic orbits. I’m suppose to check Franks and William’s paper on “Anomolous Anosov Flows” in which they gave many examples of different non-volume-preserving Anosov flows. The question of whether or not it’s possible to classify all Anosov flows (in the sense presented in the paper) is still open.

Kaufman’s construction

ConanLeave a comment

This is a note on R. Kaufman’s paper An exceptional set for Hausdorff dimension

We construct a set D \subseteq \mathbb{R}^2 with \dim(D) = d < 1 and E \subseteq [0, \pi) with \dim(E) > 0 s.t. for all directions \theta \in E, \dim(\pi_\theta(D)) < d-\epsilon (the projection of D in direction \theta is less than d-\epsilon)

\forall \alpha >1, let (n_j)_{j=1}^\infty be an rapidly increasing sequence of integers.

Define D_j = \{ (a, b)/n_j + \xi \ | \ a, b \in \mathbb{Z}, \ ||(a, b)|| \leq n_j; \ | \xi | \leq n_j^{- \alpha} \}

i.e. D_j = \bigcup \{ B((a,b)/n_j, 1/n_j^\alpha) \ | \ (a, b) \in \mathbb{Z}^2 \cap B( \overline{0}, n_j) \}

Let D = \bigcap_{j=1}^\infty D_j

\because \alpha > 1, \ (n_j) rapidly increasing, \dim(D) = 2 / \alpha

Let c \in (0, 1) be fixed, define E' = \{ t \in \mathbb{R} \ | \ \exists positive integer sequence (m_{j_i})_{i=1}^\infty s.t. m_{j_i} < C_1 n_{j_i}^c, \ || m_{j_i} t || < C_2 m_{j_i} / n_{j_i}^\alpha \}

\forall t \in E', \ \forall i \in \mathbb{N}, \ \forall p = (a, b)/n_{j_i} + \xi \in D_{j_i}, we have:

| \langle p, (1, t) \rangle - a/n_{j_i} - bt/n_{j_i} | \leq (1+|t|)/n_{j_i}^\alpha

Let b = q m_{j_i} + r where 0 \leq r < m_{j_i}, |q m_{j_i}| < C n_{j_i}

\exists z_{j_i} \in \mathbb{Z}, \ | z_{j_i} | < C | n_{j_i} |, \ | \theta |<1

bt = qm_{j_i}t +rt = X + rt + q \theta ||m_{j_i}t||

Generic Accessibility (part 1) – Andy Hammerlindl

Conan2 Comments

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…