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.

Let \phi be a smooth function supported on [a,b]\subset (0,\lambda_0). Without loss of generality we assume \int \phi dx=0. Note that \Phi(\omega,x)=\phi(x) is well defined on \Omega_\tau and \nu(\Phi)=0. We will derive a contradiction by assuming the following converges to 0 exponentially: \rho(t)=\int \Phi\cdot \Phi\circ \sigma_t d\nu.

A useful trick here is to represent \Phi(\sigma_t(\omega,x))=\sum_{n\ge0}\int_0^{\lambda(\omega_n)}\Phi(\sigma^n\omega,y)\delta_{t+x-y-\tau(\omega,n)} dy, where \delta is the \infty-pulse at 0. So the Fourier dual
\hat{\rho}(s)=\int_{\mathbb{R}}e^{ist}\rho(t)dt=\int_{\mathbb{R}} e^{ist}\int_{\Omega_\tau}\Phi\cdot \Phi\circ \sigma_t d\nu dt
=\frac{2}{\lambda_0+\lambda_1}\int_{\mathbb{R}}e^{ist}\int_{\Omega} \int_{0}^{\lambda(\omega_0)} \phi(x)\cdot \sum_{n\ge0}\int_0^{\lambda(\omega_n)}\phi(y)\delta_{t+x-y-\tau(\omega,n)} dy dx d\mu dt
=\frac{2}{\lambda_0+\lambda_1}\int_{\Omega} \int_{0}^{\lambda(\omega_0)} \phi(x)\cdot \sum_{n\ge0}\int_0^{\lambda(\omega_n)}\phi(y)(\int_{\mathbb{R}}e^{ist}\delta_{t+x-y-\tau(\omega,n)} dt) dy dx d\mu
=\frac{2}{\lambda_0+\lambda_1}\int_{\Omega} \int_{0}^{\lambda(\omega_0)} \phi(x)\cdot \sum_{n\ge0}\int_0^{\lambda(\omega_n)}\phi(y)e^{is(y+\tau(\omega,n)-x)} dy dx d\mu
=\frac{2}{\lambda_0+\lambda_1}\sum_{n\ge0}\int_{\Omega} \int_{0}^{\lambda(\omega_0)} \phi(x)e^{-isx} dx\cdot \int_0^{\lambda(\omega_n)}\phi(y)e^{isy} dy \cdot e^{is\tau(\omega,n)}  d\mu
=\frac{2\hat{\phi}(-s)\cdot \hat{\phi}(s)}{\lambda_0+\lambda_1}\sum_{n\ge0} \cdot\int_{\Omega} e^{is\tau(\omega,n)}  d\mu
=\frac{2\hat{\phi}(-s)\cdot \hat{\phi}(s)}{\lambda_0+\lambda_1} \cdot\sum_{n\ge0}\sum_{(\omega)_0^{n-1}}\prod_{0\le j<n} \frac{e^{is\lambda(\omega_j)}}{2}
=\frac{2\hat{\phi}(-s)\cdot \hat{\phi}(s)}{\lambda_0+\lambda_1}\cdot\sum_{n\ge0} \left(\frac{e^{is\lambda_0}+e^{is\lambda_1}}{2}\right)^n=\frac{2\hat{\phi}(-s)\cdot \hat{\phi}(s) }{\lambda_0+\lambda_1}\cdot\left(1-\frac{e^{is\lambda_0}+e^{is\lambda_1}}{2}\right)^{-1}.

Remark 1. Ruelle’s assumption on \phi simplifies the expression greatly: \hat{\phi}(s)=\int_{0}^{\lambda(\omega_j)} \phi(x)e^{isx} dx since \phi is supported on [a,b]\subset (0,\lambda_0). Moreover as pointed out by Ruelle, there are poles of \left(1-\frac{e^{is\lambda_0}+e^{is\lambda_1}}{2}\right)^{-1} near the real axis and it may be arranged that these are not zeros of \hat{\phi}(\pm\bullet). So \rho(t) does not decay exponentially.

Remark 2. Now let’s embed \Omega into a diffeomorphism (M,f) such that f|_{\Omega}=\sigma. Clearly two cylinders [0] and [1] are 3\delta-separated for some \delta>0. Assume the suspension function is also extended to \tau:M\to[\lambda_0,\lambda_1] with B([i],\delta)\subset\tau^{-1}(\lambda_i). Then we see that the local stable manifold W^s_\delta(x,f) gets embedded to the local stable manifold W^s_\delta((x,t),f_{\bullet}), so is the local unstable manifolds . In particular E^s\oplus E^u is locally integrable at \Omega_\tau (although \Omega_\tau may not be open). So integrability may be one of the obstructions for exponential decay. In fact uniform non-integrability condition is the key assumption of Dolgolpyat’s paper here for the exponential decay of SRB measures of Anosov flows with C^1 foliations.

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