Abstract. Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen's entropy for amenable group action dynamical systems and show that under the tempered condition, the Bowen entropy of the whole compact space for a given Følner sequence equals to the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin-Katok's local entropy formula for dynamical systems with amenable group actions.
Let X be a compact metric space, f a continuous transformation on X, and Y a vector space with linear compatible metric. Denote by M(X) the collection of all the probability measures on X. For a positive integer n, define the nth empirical measure L n : X → M(X) aswhere δ x denotes the Dirac measure at x. Suppose : M(X) → Y is continuous and affine with respect to the weak topology on M(X). We think of the compositeas a continuous and affine deformation of the empirical measure L n . The set of divergence points of such a deformation is defined as D(f, ) = {x ∈ X | the limit of L n x does not exist}.In this paper we show that for a continuous transformation satisfying the specification property, if (M(X)) is a singleton, then set of divergence points is empty, i.e. D(f, ) = ∅, and if (M(X)) is not a singleton, then the set of divergence points has full topological entropy, i.e. h top (D(f, )) = h top (f ).
In this article, the historic set is divided into different level sets and we use topological pressure to describe the size of these level sets. We give an application of these results to dimension theory. Especially, we use topological pressure to describe the relative multifractal spectrum of ergodic averages and give a positive answer to the conjecture posed by L. Olsen (J. Math. Pures Appl. 82 (2003)).
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