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

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.