Entropy, Pressure And Large Deviations For Amenable Group Action Dynamical Systems

In this paper, we will focus on the Bowen topological entropy and large devia-tions for amenable group action dynamical systems.We first introduce the definition of Bowen topological entropy for compact met-ric amenable group action systems. By using ideas from geometric measure theory,we establish the following variational principle via Bowen topological entropy for amenable group actions:Bowen topological entropy on a compact subset equals to the supremum of the(lower) local entropy with respect to the Borel probability measures that support on this set.We also obtain some formulas related to the local entropy. As applications, we prove that:1. Bowen topological entropy of the whole system equals to the usual topo-logical entropy defined by open covers.2. If ? is an invariant Borel probability measure and Y a subset with ?(Y)=1, then the Bowen topological entropy of Y is bigger than the measure-theoretical entropy with respect to ?.3. If in addition ? is ergodic, then the Bowen topological entropy of the set of generic points of ? is equal to the measure-theoretical entropy with respect to ?.The above results generalize the classical results of Bowen in 1973 to the amenable group action framework.Moreover, we consider the exponential growth rate for the measure of some set-s via ergodic average and obtain the corresponding large deviation theorems for a-menable group actions.