Entropies of a semigroup of maps

On the topological entropy of a semigroup of continuous maps

“…Thus the new definition of topological entropy of G is the same as the topological entropy of G defined by Biś [2]. We write H d (G 1 ), H d (G 1 , K) respectively to emphasize d if we need to.…”

“…A subset E of X is called (n, )-spanning set of X if for every x ∈ X there exists y ∈ E such that d n (x, y) ≤ . Let r(n, , X, G 1 ) denote the smallest cardinality of any (n, )-spanning sets of X. Biś [2] showed that for any semigroup G generated by a finite set G 1 the following equality holds h(G 1 ) = lim →0 lim sup n→∞ 1 n log(r(n. , X, G 1 )).…”

“…Many useful properties of the concept of entropy of a single transformation can be found in [22]. In the following, we will give some estimates of the topological entropy given in Biś [2] and the new topological entropy given in Section 3.…”

“…Ghys, Langevin, and Walczak in [9] proposed a definition of a topological entropy for finitely generated pseudogroups of continuous transformations. Biś [2] introduced the notion of topological entropy of a semigroup of finite continuous maps. Moreover, Biś and Urbański in [3] investigate the notion of topological entropy of a semigroup of continuous maps and provide several of its basic properties.…”

An analogue of the Variational Principle for group and pseudogroup actions

“…However, a few different definitions of entropy of a semigroup are known ( [11], [8], [26], [3], [4]) and most of them are unrelated. For example, both Bufetov in [8] and Sumi in [26] apply the idea of skewproduct transformations.…”