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Discrete dynamical systems in group theory. (English) Zbl 1280.37023
In this very interesting expository article, a unifying categorial approach for several dynamical entropy functions, defined in different categories of discrete dynamical systems, is introduced. The main idea is to define a universal entropy function \(h_{\mathfrak{S}}:\mathrm{Flow}_{\mathfrak{S}}\rightarrow[0,\infty]\) on the category \(\mathrm{Flow}_{\mathfrak{S}}\) of contractive endomorphisms of normed semigroups. For a normed semigroup \((S,\cdot,v)\) and a contractive endomorphism \(\phi:S\rightarrow S\) one defines \[ T_n(\phi,x) = x \cdot \phi(x) \cdot \ldots \cdot \phi^{n-1}(x) \text{\quad and \quad } c_n(\phi,x) := v(T_n(\phi,x)) \] for \(x\in S\) and \(n\in\mathbb{N}\). Then the semigroup entropy of \(\phi\) is \[ h_{\mathfrak{S}}(\phi) := \sup_{x\in S}\lim_{n\rightarrow\infty}\frac{1}{n}c_n(\phi,x). \] This entropy function satisfies the elementary properties that one usually associates with a dynamical entropy, as for instance invariance under conjugation, invariance under inversion, a power rule, and monotonicity with respect to restrictions. The general idea how to obtain the known entropy functions (topological entropy or metric entropy, for instance) is to set \[ \qquad h_F:\mathrm{Flow}_{\mathfrak{X}}\rightarrow[0,\infty],\quad h_F(\phi) := h_{\mathfrak{S}}(F(\phi)), \] where \(\mathfrak{X}\) is the corresponding category (compact topological spaces with continuous maps or probability spaces with measure-preserving maps, for instance) and \(F:\mathfrak{X}\rightarrow\mathfrak{S}\) is an appropriate functor into the category of normed semigroups. Depending on the specific properties of the functor \(F\), \(h_F\) inherits different properties of \(h_{\mathfrak{S}}\). In the case of topological entropy, for instance, the functor assigns to a compact topological space \(X\) the normed semigroup of open covers of \(X\) with the join operation and the norm given by the logarithm of the minimal subcover cardinality, and to a continuous map its induced map on open covers. Also different notions of algebraic entropy for group endomorphisms can be obtained via this general scheme. In particular, this allows the authors to give a new proof of the Weiss Bridge theorem [M. D. Weiss, Math. Syst. Theory 8(1974), 243–248 (1975; Zbl 0298.28014)], and other Bridge theorems, relating entropy functions in different categories. Finally, a notion of growth for dynamical systems in the category of groups is introduced, which is an extension of the classical notion of growth for groups.

37B40 Topological entropy
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
20M15 Mappings of semigroups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20F65 Geometric group theory
22D35 Duality theorems for locally compact groups
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
54C70 Entropy in general topology
55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
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