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Algebraic flows and their entropy Dikran Dikranjan

Summary: Algebraic flows and their entropy
Dikran Dikranjan
Dipartimento di Matematica e Informatica, Universit`a di Udine,
Via delle Scienze, 206 - 33100 Udine, Italy
A flow in a category X is an object X of X provided with an endomorphism T : X X in X. In most of the
cases X will be the category of (topological) groups and (continuous) group homomorphisms, the category of right
modules over a ring R and the R-module homomorphisms, or just the category of topological (resp., measure) spaces
and continuous (resp., measure preserving) maps. An isomorphism between two flows T : X X and S : Y Y
is an isomorphism : X Y in X such that -1
S = T.
A fundamental numerical invariant used to classify the flows up to isomorphism is the entropy. It was introduced
in ergodic theory by Kolmogorov and Sinai in 1958, and in topological dynamics by Adler, Konheim, and McAndrew
[1]. These authors proposed also a brief general scheme for defining algebraic entropy in the context of abelian
groups, developed further in [11, 4]. Since this approach was appropriate only for torsion groups, a modification
was proposed by Peters [7] in the case of non-torsion abelian groups. A second modification was proposed in [2],
since Peters' approach works only for monomorphisms. Adjoint (dual) entropy in abelian groups was introduced in
[3, 6]. The algebraic entropy was extended to the context of modules by Salce and Zanardo [9]. In all these cases
the entropy is intended to measure the "chaos" or "disorder" created by the discrete dynamical system. Recently
Salce, Vamos and Virili [8] found a fruitful connection between the algebraic entropy of module endomorphisms and


Source: Angeleri Hügel, Lidia - Dipartimento di Informatica, Universita` di Verona


Collections: Mathematics