Algebra, Logic and Topology


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The Tale of Three Entropies

To evaluate the “chaos” or “disorder” caused by a transformation T : K-->K (preserving the natural structure of K) one defines the entropy of T. The algebraic entropy of endomorphisms T : K-->K of an abelian group K was introduced by Adler, Konheim and McAndrew. In the same paper they introduced also the topological entropy of the continuous self-maps of compact topological spaces. The specific case of compact topological groups K and their endomorphisms T : K-->K is of special interest by the fact, first noticed by Paul Halmos, that the surjective endomorphisms are also measure preserving maps with respect to the Haar measure of the group K. Hence the measure-theoretic entropy of T can be also considered and it turns out to coincide with the topological one. In case K is abelian, one can consider also the discrete Pontryagin dual group bK and the adjoint endomorphism bT : bK-->bK. It certain cases (e.g., when K is pro-finite, or metrizable [i.e., when bK is torsion, or countable]), the topological entropy of T coincides also with the algebraic entropy of bT (according to theorems of Weiss and Peters, resp.). In other words, under appropriate conditions all three entropies agree. The talk is addressed to a general audience and will discuss some recent results on the topological entropy of endomorphisms of compact abelian groups and the algebraic entropy of the endomorphisms of Abelian groups.

Dikran Dikranjan (Udine University, Italy)

October 30, 2007


Sala 5.5


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