Tag: hyperbolicity

A remark on a mini-course by Kleiner in Sullivan’s 70th birthday

Conan2 Comments

I spent the last week on Long Island for Dennis Sullivan’s birthday conference. The conference is hosted in the brand new Simons center where great food is served everyday in the cafe (I think life-wise it’s a wonderful choice for doing a post-doc).

Anyways, aside from getting to know this super-cool person named Dennis, the talks there were interesting~ There are many things I found so exciting and can’t help to not say a few words about, however due to my laziness, I can only select one item to give a little stupid remark on:

So Bruce Kleiner gave a 3-lecture mini-course on boundaries of Gromov hyperbolic spaces (see this related post on a piece of his pervious work in the subject)

Cannon’s conjecture: Any Gromov hyperbolic group with \partial_\infty G \approx \mathbb{S}^2 acts discretely and cocompactly by isometries on \mathbb{H}^3.

As we all know, in the theory of Gromov hyperbolic spaces, we have the basic theorem that says if a groups acts on a space discretely and cocompactly by isometries, then the group (equipped with any word metric on its Cayley graph) is quasi-isometric to the space it acts on.

Since I borrowed professor Sullivan as an excuse for writing this post, let’s also state a partial converse of this theorem (which is more in the line of Cannon’s conjecture):

Theorem: (Sullivan, Gromov, Cannon-Swenson)
For G finitely generated, if G is quasi-isometric to \mathbb{H}^n for some n \geq 3, then G acts on \mathbb{H}^n discretely cocompactly by isometries.

This essentially says that due to the strong symmetries and hyperbolicity of \mathbb{H}^n, in this case quasi-isometry is enough to guarantee an action. (Such thing is of course not true in general, for example any finite group is quasi-isometric to any compact metric space, there’s no way such action exists.) In some sense being quasi-isometric is a much stronger condition once the spaces has large growth at infinity.

In light of the above two theorems we know that Cannon’s conjecture is equivalent to saying that any hyperbolic group with boundary \mathbb{S}^2 is quasi-isometric to \mathbb{H}^3.

At first glance this seems striking since knowing only the topology of the boundary and the fact that it’s hyperbolic, we need to conclude what the whole group looks like geometrically. However, the pervious post on one dimensional boundaries perhaps gives us some hint on the boundary can’t be anything we want. In fact it’s rather rigid due to the large symmetries of our hyperbolic group structure.

Having Cannon’s conjecture as a Holy Grail, they developed tools that give raise to some very elegant and inspring proofs of the conjecture in various special cases. For example:

Definition: A metric space M, is said to be Alfors \alpha-regular where \alpha is its Hausdorff dimension, if there exists constant C s.t. for any ball B(p, R) with R \leq \mbox{Diam}(M), we have:

C^{-1}R^\alpha \leq \mu(B(p,R)) \leq C R^\alpha

This is saying it’s of Hausdorff dimension \alpha in a very strong sense. (i.e. the Hausdorff \alpha measure behaves exactly like the regular Eculidean measure everywhere and in all scales).

For two disjoint continua C_1, C_2 in M, let \Gamma(C_1, C_2) denote the set of rectifiable curves connecting C_1 to C_2. For any density function \rho: M \rightarrow \mathbb{R}^+, we define the \rho-distance between C_1, C_2 to be \displaystyle \mbox{dist}_\rho(C_1, C_2) = \inf_{\gamma \in \Gamma(C_1, C_2)} \int_\gamma \rho.

Definition: The \alpha-modulus between C_1, C_2 is

\mbox{Mod}_\alpha(C_1, C_2) = \inf \{ \int_M \rho^\alpha \ | \ \mbox{dist}_\rho(C_1, C_2) \geq 1 \},

OK…I know this is a lot of seemingly random definitions to digest, let’s pause a little bit: Given two continua in our favorite \mathbb{R}^n, new we are of course Hausdorff dimension n, what’s the n-modulus between them?

This is equivalent to asking for a density function for scaling the metric so that the total n-dimensional volume of \mathbb{R}^n is as small as possible but yet the length of any curve connecting C_1, \ C_2 is larger than 1.

So intuitively we want to put large density between the sets whenever they are close together. Since we are integrating the n-th power for volume (suppose n>1, since our set is path connected it’s dimension is at least 1), we would want the density as ‘spread out’ as possible while keeping the arc-length property. Hence one observation is this modulus depends on the pair of closest points and the diameter of the sets.

The relative distance between C_1, C_2 is \displaystyle \Delta (C_1, C_2) = \frac{\inf \{ d(p_1, p_2) \ | \ p_1 \in C_1, \ p_2 \in C_2 \} }{ \min \{ \mbox{Diam}(C_1), \mbox{Diam}(C_2) \} }

We say M is \alpha-Loewner if the \alpha modulus between any two continua is controlled above and below by their relative distance, i.e. there exists increasing functions \phi, \psi: [0, \infty) \rightarrow [0, \infty) s.t. for all C_1, C_2,

\phi(\Delta(C_1, C_2)) \leq \mbox{Mod}_\alpha(C_1, C_2) \leq \psi(\Delta(C_1, C_2))

Those spaces are, in some sense, regular with respect to it’s metric and measure.

Theorem: If \partial_\infty G is Alfors 2-regular and 2-Loewner, homeomorphic to \mathbb{S}^2, then G acts discrete cocompactly on \mathbb{H}^3 by isometries.

Most of the material appeared in the talk can be found in their paper.

There are many other talks I found very interesting, especially that of Kenneth Bromberg, Mario Bonk and Peter Jones. Unfortunately I had to miss Curt McMullen, Yair Minski and Shishikura…

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