## 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 $\sigma_t$.

### New notes

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. $x$.

### 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.