On amenable semigroups of rational functions

TL;DR

This paper characterizes when semigroups of rational functions are amenable under composition, showing they are often subsemigroups of the centralizer of a rational function, with specific results for polynomials.

Contribution

It provides a comprehensive characterization of amenable semigroups of rational functions and polynomials, linking amenability to centralizers of rational functions.

Findings

Semigroups of polynomials are characterized as amenable under composition.

Amenable semigroups of rational functions are subsemigroups of the centralizer of some rational function.

Results apply under quite general conditions for rational functions.

Abstract

We characterize left and right amenable semigroups of polynomials of one complex variable with respect to the composition operation. We also prove a number of results about amenable semigroups of arbitrary rational functions. In particular, we show that under quite general conditions a semigroup of rational functions is left amenable if and only if it is a subsemigroup of the centralizer of some rational function.