hyperbolic dynamics

1. Anosov Closing Lemma. If \Lambda is a hyperbolic set for a diffeomorphism (M,f), then there exists a K\ge1 such that given any sufficiently small \epsilon>0 and a \epsilon-pseudo-orbit \{x_i\}_{i=0}^n in \Lambda with x_n=x_0, there is an n-periodic point x such that d(x_i,f^ix)\le K\epsilon for each 0\le i\le n.

This also indicates that any Anosov diffeomorphism for which M itself is hyperbolic, is Axiom A: \Omega(f) is hyperbolic and \mathrm{Per}(f) is dense \Omega(f).

2. Specification Property. Given any \epsilon>0 there is a relaxation time N_\epsilon such that every N_\epsilon-spaced collection of orbit segments is \epsilon-shadowed by an actual/genuine orbit. Moreover, one can choose the shadowing point to be a periodic point with period no more than T+M.

Note that the time between the segments depends only on the quality of the approximation and not on the length of the specified segments. Bowen’s Specification Theorem says that compact topologically transitive hyperbolic sets have Specification Property. (e.g. the basic sets of the nonwandering set of Axiom A diffeomorphism)

It is proved that for a system with specification property, each invariant measure has some generic point. In this case for every continuous potential \phi:X\rightarrow\mathbb{R}, the Lyapunov spectrum of the potential \mathfrak{L}_\phi=\{a\in\mathbb{R}|\frac{1}{n}\sum_{0\le k<n}\phi(f^kx)\rightarrow a\text{ for some }x\in X\} is a closed interval.

3. Nonuniform hyperbolicity. Assume f has Holder derivative. Let x be a Perron regular point. Then the stable sest W^s(x)=\{y\in M :d(f^ny,f^nx)\to 0\} is a smooth immersed disk tangent to E^s_x (hence the name stable manifold). It can be generated as W^s(x)=\bigcup_{n\ge0}f^{-n}W^s_{loc}(f^nx).

Let \mathcal{H}_k be the Pesin hyperbolic Block for k\ge1. Then W^s_{loc}(x) varies continuously with x\in \mathcal{H}_k. In particular they have uniform size, so are the local unstable manifolds and the angle between them. The collection \mathcal{W}^s=\{W^s_{loc}(x):x\in \mathcal{H}_k\} for a lamination over \mathcal{H}_k.

Let x\in\mathcal{H}_k and p,q\in W^s_{loc}(x) close to x. Let \Sigma_p be small smooth disks transverse to W^s_{loc}(x) at p, similar is \Sigma_q. For each y\in\mathcal{H}_k close to x, W^s_{loc}(y) intersects each transverse in exactly one point y(p/q). This induces a holonomy map
h_s:A\subset\Sigma_p\to B\subset\Sigma_q,  y(p)\mapsto y(q).
This is a homeomorphism between A and B.
Pesin theorem: the holonomy map h_s is absolutely continuous: it maps zero Lebesgue measure subsets of A to zero Lebesgue measure subsets of B, so is h_s^{-1}. In words, the stable lamination \mathcal{W}^s is absolutely continuous.

4. Direct argument for the following fact.
Let f\in\mathrm{Diff}^1M be a diffeo on a compact manifold M and \mu be an f-invariant and ergodic measure with only strictly negative Lyapunov exponents: \chi^i_\mu<0 for all i. Then \mu is carried by a periodic orbit.

For example let p be a attracting periodic point. Then the average \frac{1}{|\mathcal{O}(p)|}\sum_{y\in\mathcal{O}(p)}\delta_{y} is an measure satisfying above assumption.

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Comments

  • 777  On February 16, 2010 at 5:34 am

    In the first paragraph, I think you meant given any epsilon > 0 and an \epsilon-pseudo orbit. (you missed the “\epsilon” before ‘pseudo orbit’ as otherwise multiplying constant K does not make sense)

    Nice blog~^^

    • 777  On February 16, 2010 at 11:56 pm

      Haha…Back to that question you asked some time ago…>.<
      Just to clarify, your setting is, for M Riemannian manifold; f:M \rightarrow M ergodic w.r.t.\mu and for all v \in TM the Lyapunov exponents is neigative?

      If that's the right setting, then can the support even be a periodic orbit? (somehow I can only imaging the support being a fixed point)

      Can we just do the following: take any two point p, q, join them by a minimal geodesic, consider then length of the image of the geodesic as a curve from $f^n(p)$ to $f^n(q)$ by integration the length of the curve tends to 0 as $n \rightarrow \infty$ so the distance between any two points is tending to 0 with an contraction constant at least c<1 so diameter of the support should be 0.

      I must have misunderstood something about your question…

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