Abstract
Abstract The usual notion of algebraic entropy associates to every group (monoid) endomorphism a value estimating the chaos created by the self-map. In this paper, we study the extension of this notion to arbitrary sets endowed with monoid actions, providing properties and relating it with other entropy notions. In particular, we focus our attention on the relationship with the coarse entropy of bornologous self-maps of quasi-coarse spaces. While studying the connection, an extension of a classification result due to Protasov is provided.
Highlights
After Clausius’ de nition in thermodynamics in 1865, entropy in mathematics was rstly introduced by Shannon in information theory ([32])
We study the extension of this notion to arbitrary sets endowed with monoid actions, providing properties and relating it with other entropy notions
A di erent approach was provided by Peters in [23], and it was generalised for all endomorphisms of abelian groups in [9]
Summary
After Clausius’ de nition in thermodynamics in 1865, entropy in mathematics was rstly introduced by Shannon in information theory ([32]). (c) Let M be a (left-)cancellative commutative monoid, G(M) be the abelian group generated by M ([5]), and ı : M → G(M) the inclusion homomorphism. Consider the group X = Z = Y = Z = { , }, endowed with its right regular action, and the homomorphisms h = idX and g(G) = { }. Let us suppose that for every pair of distinct points of Z, their orbits are disjoint and the homomorphism g satis es the following property: for every a, b ∈ M, αN(g(a)) = αN(g(b)) provided that there exists x ∈ X such that αM(a)(x) = αM(b)(x). Let M be a left-cancellative monoid endowed with its right regular action, and (f , f ) and (f , f )
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