Bowen extended Sinai’s construction of Markov partition to Axiom A diffeomorphism, and proved that the mixing rate of every Gibbs measure with respect to a Holder potential over a mixing basic set is always exponential, that is,

for all smooth functions on , the correlations 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 and be the shift on . Pick two positive numbers such that is irrational. Define a ceiling function . Then let be the suspension flow over with respect to . It is well known that there is a one to one correspondence between the -invariant measures and -invariant measures : . Ruelle examined the measure of maximal entropy, (corresponding the zero potential) and showed that the corresponding measure does not mix exponentially under .

Let be a smooth function supported on . Without loss of generality we assume . Note that is well defined on and . We will derive a contradiction by assuming the following converges to 0 exponentially: .

A useful trick here is to represent , where is the -pulse at . So the Fourier dual

.

**Remark 1. **Ruelle’s assumption on simplifies the expression greatly: since is supported on . Moreover as pointed out by Ruelle, there are poles of near the real axis and it may be arranged that these are not zeros of . So does not decay exponentially.

**Remark 2. ** Now let’s embed into a diffeomorphism such that . Clearly two cylinders and are -separated for some . Assume the suspension function is also extended to with . Then we see that the local stable manifold gets embedded to the local stable manifold , so is the local unstable manifolds . In particular is locally integrable at (although 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 foliations.