  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.
