Tag Archives: exponential dacay

Mixing but non-exponentially mixing Axiom A flows: Ruelle’s example

Bowen extended Sinai’s construction of Markov partition to Axiom A diffeomorphism, and proved that the mixing rate of every Gibbs measure \mu with respect to a Holder potential over a mixing basic set is always exponential, that is,
for all smooth functions \phi,\psi on M, the correlations \rho(t)=\int \phi\cdot\psi\circ \sigma^n d\mu-\mu(\phi)\cdot\mu(\psi)\to 0 exponentially.
However the situation is quite different for Axiom A flows. Ruelle gave the first class of examples: mixing but non-exponentially mixing Axiom A flows.

Ruelle first constructed a symbolic example and then embedded it into an Axiom A flow. Let \Omega=\{0,1\}^{\mathbb{Z}} and \sigma be the shift on \Omega. Pick two positive numbers \lambda_0<\lambda_1 such that \frac{\lambda_0}{\lambda_0} is irrational. Define a ceiling function \tau:\omega\in\Omega\mapsto \lambda(\omega_0). Then let (\Omega_\tau,\sigma_t) be the suspension flow over (\Omega,\sigma) with respect to \tau. It is well known that there is a one to one correspondence between the \sigma_t-invariant measures \nu and \sigma-invariant measures \mu: d\nu=\frac{dt\times d\mu}{\mu(\tau)}. Ruelle examined the measure of maximal entropy, \mu (corresponding the zero potential) and showed that the corresponding measure \nu does not mix exponentially under \sigma_t.

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4. Ledrappier-Young’s criterion for measures with Absolutely Continuous Conditional measures on Unstable manifolds. Let f:M\to M be a C^2 diffeomorphism and \mu be an invariant probability measure. For \mu-a.e. x, if there exist negative Lyapunov exponent(s) at x, then the set of points with exponentially approximating future of x is a C^2-submanifold, W^u(x). Then \mu is said to have ACCU if for each measurable partition \xi with \xi(x)\subset W^u(x) and contains an unstable plaque for \mu-a.e. x, the conditional measure \mu_{\xi(x)}\ll m_{W^u(x)} for \mu-a.e. x.

Theorem. \mu has ACCU if and only if h_\mu(f)=\Lambda^+(\mu), where \Lambda^+(\mu)=\int\sum\lambda^+_i(x)d\mu(x).

Definition. Such a measure is called the SRB measure.

Moreover, the density \frac{d\mu_{\xi(x)}}{dm_{W^u(x)}} is strictly positive and C^1 on \xi(x) for \mu-a.e. x.

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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.