Discrete dynamical systems in group theory

In this expository paper we describe the unifying approach for many known entropies in Mathematics developed in [27]. First we give the notion of semigroup entropy h(G) : G -> R+ in the category G of normed semigroups and contractive homomorphisms, recalling also its properties from [27]. For a specific category X and a functor F : X -> G we have the entropy F : X -> G, defined by the composition h(F) : h(G)degrees F, which automatically satisfies the same properties proved for h(G). This general scheme permits to obtain many of the known entropies as h(F), for appropriately chosen categories X and functors F :X -> G . In the last part we recall the definition and the fundamental properties of the algebraic entropy for group endomorphisms, noting how its deeper properties depend on the specific setting. Finally we discuss the notion of growth for flows of groups, comparing it with the classical notion of growth for finitely generated groups.