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.
(Udine University, Italy)
October 30, 2007