Topological and algebraic entropy of actions of amenable cancellative monoids

The notion of topological entropy was first introduced by Adler, Konheim and McAn- drew in 1965, as an invariant of continuous self-maps of compact Hausdorff topological spaces. Few years later, Bowen introduced a notion of entropy for uniformly continuous self-maps of metric spaces, later extended to general uniform spaces by Hood. On a com- pact topological group, endowed with its unique compatible uniformity, all these notions of entropy take the same value, which also coincides with the measure entropy, widely used in information theory, with respect to the Haar measure.
In the same paper where they introduced the topological entropy, Adler, Konheim and McAndrew also proposed a dual notion, later called algebraic entropy, that applies to endomorphisms of discrete groups. It was then shown by Peters that the algebraic entropy of an endomorphism coincides with the topological entropy of the continuous endomophism induced on the Pontryagin-Van Kampen dual.
Both the theory of algebraic and of topological entropy have been extended first to actions of amenable groups and, more recently, even to actions of sofic groups, with important contributions by Ornstein, Weiss, Bowen, Li, Kerr and others.
In this talk, we will show how to extend both the topological and the algebraic entropy to include actions of right amenable and cancellative monoids (on compact Hausforff spaces and on discrete Abelian groups, respectively). In particular, we will prove that any right amenable and cancellative monoid is left Ore and we will use the theory of modules of fractions to reduce the computation of the algebraic entropy of actions of such a monoid, to that of an action of its amenable group of left fractions.
A close analysis of the construction of modules of fractions allows us to formally dualize this process and find a functorial procedure that, to an action of a right amenable and cancellative monoid on a compact Hausdorff group, associates an action of the amenable group of left fractions. Furthermore, the topological entropy of the two actions remains the same.
This is a joint work with D. Dikranjan and A. Giordano Bruno.