Tag Archives: Anosov Closing Lemma

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.