2013/08/19 by Dikran Dikranjan, Dikranjan, Dikran, Anna Giordano Bruno +1
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1308.4035
In this expository paper we describe an unifying approach for many known entropies in Mathematics. First we recall the notion of semigroup entropy hS in the category S of normed semigroups and contractive homomorphisms, recalling also its properties. For a specific category X and a functor F from X to S, we have the entropy hF, defined by the composition of hS with F, which automatically satisfies the same properties proved for hS. This general scheme permits to obtain many of the known entropies as hF, for appropriately chosen categories X and functors F. 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.