Author: Conan

Outcast, hippie, ex-mathematician, ex-theme park painter, ex-management consultant, ex-software engineer & generally someone who refuses to settle...

Anosov flows

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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.

Types of hyperbolicity

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Axiom A

1. Nonwandering set is hyperbolic

2. Periodic points are dense in the nonwandering set

Kupka-Smale

1. All periodic points are hyperbolic

2. For each pair of periodic points p, q of f, the intersection between the stable manifold of $p$ and the unstable manifold of q is transversal

Kupka-Smale theorem

The set of Kupka-Smale diffeomorphisms is residual in \mbox{Diff}^r(M) under C^r topology.

Morse-Smale

1.Axiom A with only finitely many periodic points (hence \Omega(f) is just the set of periodic points)

2.For each pair of periodic points p, q of f, the intersection between the stable manifold of p and the unstable manifold of q is transversal.

Anosov

All points are hyperbolic, i.e. there is a splitting of the whole tangent bundle such that under the diffeo, stable directions are exponentially contracted and unstable directions are exponentially expanded.

Relations:

Morse-Smale \subseteq Axiom A

Morse-Smale \subseteq Kupka-Smale

Anosov \subseteq Axiom A

Kaufman’s construction

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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||

A question by Furstenberg

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Yesterday I was talking about some properties of different dimensions with Furstenburg. Somehow I mentioned Kekaya, and he told me about the following question he has been longing to solve (which is amazingly many similarities to Kekaya):

For set S \in \mathbb{R}^2, if \exists \delta>0 s.t. for all direction \theta, \exists line l with direction \theta s.t. \dim (l \cap S) > \delta . Does it follow that \dim(A) \geq 1 ?

Note that a stronger conjecture would be \dim(A) is at least 1+\delta which when taking \delta = 1 would give a generalization of the 2-dimensional Kekaya. (i.e. instead of having to have a line segment, we only require a 1-dimension set in each direction)

Reviewing the proofs of the 2-dimsional Kekaya, I found they all rely on the fact that the line segment is connected…Hence it might be interesting to even find an answer to the following question:

If A \subseteq \mathbb{R}^2 contains a measure 1 set in every direction, does it follow that \dim(A)=2?

Billiards

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Amie Wilkinson asked me the following question some time ago:

Given a smooth convex Jordan curve J \subseteq \mathbb{R}^2, consider the billiard map \varphi: J \times (0, \pi) \rightarrow J \times (0, \pi), let \pi_\theta: J \times (0, \pi) \rightarrow (0, \pi) be the projection.

a) If \forall (p, \theta) \in J \times (0, \pi), \pi_\theta \circ \varphi (p, \theta) = \theta, does this imply J is a circle?

(Yes, in fact we only need \varphi to fix the second component of points in \{p, q \} \times (0, \pi) for a chosen pair of points p, q. Classical geometry)

b) If \forall p \in J, \pi_\theta \circ \varphi (p, \pi/2) = \pi/2, does this imply J is a circle?

(No, my example was a cute construction that attaches six circular arcs together)

c) What’s the smallest set S \subseteq (0, \pi) s.t. if \varphi fixes the second component on J \times S then J has to be a circle?

I am still thinking about c)…My guess is that any sub interval would work, and of course any dense subset inside a given set works equally well as the whole set…

But is it possible to have only finitely many angles? Maybe even two angles?