Speaker: Prof. Dikran Dikranjan, Department of Mathematics and Computer Science, University of Udine
Title: Topological Entropy of Endomorphisms of Compact Abelian Groups or the Tale of Three Entropies
Abstract: 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 K^ and the
adjoint endomorphism T^: K^ →
K^. It certain cases (e.g., when K is pro-finite, or
metrizable [i.e., when K^ is torsion, or countable]), the
topological entropy of T coincides also with the algebraic entropy of
T^ (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.