Tag Archives: nonuniform hyperbolicity

nonuniform conditions and uniform conclusions

Let f:X\to X be a continuous map. CAO Yongluo proved that if \psi\in C(X,\mathbb{R}) is continous and \mu(\psi) < \lambda for all \mu\in\mathcal{M}(f), then
for EVERY x\in X, there exists n_x\ge 1 such that the Birkhoff average \psi_f(x,n_x) < \lambda;
Moreover there exists a uniform N\ge1 such that \psi_f(x,n) < \lambda for every x\in X and n\ge N.

Now let \phi_n: X\to \mathbb{R} be a family of continuous subadditive potential with respect to (X,f). By subadditive ergodic theorem we know that the limit \phi_f(x)=\lim_{n\to\infty} \frac{\phi_n(x)}{n} exists on a set of full probability.

CAO: If \phi_f(x) < 0 on a set of full probability, then there exists a uniform L\ge 1 and \lambda < 0 such that
\int \frac{\phi_N}{N} d\mu < \lambda for every \mu\in \mathcal{M}(f).

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Anderson Localizaion

Today Zhenghe explained Anderson Localizaion to me.
(update according to Zhenghe’s comment)

Consider a Schrodinger operator H_{f,\theta}:\ell^2(\mathbb{Z})\to\ell^2(\mathbb{Z}). Let \Sigma_{pp}(H_{f,\theta})\overset{\triangle}{=}\overline{EV(H_{f,\theta})} and \ell^2(\mathbb{Z})_{pp} be the subspace spanned by all eigenvectors of E\in EV(H_{f,\theta}).

The Schrodinger operator H_{f,\theta} is said to displays Anderson Localization if

1. the eigenvectors span the whole space: \ell^2(\mathbb{Z})=\ell^2(\mathbb{Z})_{pp} (evidently this is stronger than \Sigma(H_{f,\theta})=\Sigma_{pp}(H_{f,\theta})).

2. for each E\in EV(H_{f,\theta}) with the nontrivial eigenvector \phi_E\in\ell^2(\mathbb{Z}) there exist \epsilon>0, C\ge1 and n_E\in\mathbb{Z} such that |\phi_E(n)|\le~C\cdot e^{-\epsilon |n-n_E|} for each n\in \mathbb{Z}.

Note that for SL(2,\mathbb{R}) cocycle, if an e.v. E has some \phi which decays exponentially, then this e.v. E must be simple.

It is quite interesting. The exponential decayed eigenvector corresponds to an orbit of a Schrodinger cocycle. Hence there exists a vector that is exponentially contracted under both forward and backward iterates. This resembles the case that a vector lies in the homoclinic tangency of a hyperbolic fixed point. This is an obstacle of uniform hyperbolicity. So positive Lyapunov exponent could only coexists with nonunifom hyperbolicity.

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.