On amenability and measure of maximal entropy for semigroups of rational maps: II

TL;DR

This paper explores the algebraic and dynamical properties of semigroups of rational maps, establishing connections with classical conjectures and problems in dynamics and number theory, and providing partial solutions and new insights.

Contribution

It proves a version of the Day-von Neumann's conjecture, offers a partial answer to Sushkievich's problem, and relates these to Furstenberg's $ imes 2 imes 3$ problem for semigroups of rational maps.

Findings

Proves a version of the Day-von Neumann's conjecture for semigroups of rational maps.

Provides a partial positive answer to Sushkievich's problem.

Establishes a connection between these conjectures and Furstenberg's $ imes 2 imes 3$ problem.

Abstract

We compare dynamical and algebraic properties of semigroups of rational maps. In particular, we show a version of the Day-von Neumann's conjecture and give a partial positive answer to "Sushkievich's problem" for semigroups of rational maps. We also show the relation of these conjectures with Furstenberg's $\times 2 \times 3$ problem and prove a coarse version of Furstenberg's problem for semigroups of non-exceptional polynomials.