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 .