# On amenable semigroups of rational functions

@article{Pakovich2020OnAS, title={On amenable semigroups of rational functions}, author={Fedor Pakovich}, journal={arXiv: Dynamical Systems}, year={2020} }

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.

## 7 Citations

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

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This article is a continuation of [5] and gathers some of recent developments in semigroups of rational maps. In particular, we discuss a version of Day-von Neumann conjecture and Sushkievich…

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We formulate some problems and conjectures about semigroups of rational functions under composition. The considered problems arise in different contexts, but most of them are united by a certain…

On iterates of rational functions with maximal number of critical values

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Let $F$ be a rational function of one complex variable of degree $m\geq 2$. The function $F$ is called simple if for each $z\in \mathbb C\mathbb P^1$ the preimage $P^{-1}\{z\}$ contains at least…

Lower bounds for genera of fiber products

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We provide lower bounds for genera of components of fiber products of holomorphic maps between compact Riemann surfaces.

A Tits alternative for rational functions

- Mathematics
- 2021

We prove an analog of the Tits alternative for rational functions. In particular, we show that if S is a finitely generated semigroup of rational functions over C, then either S has polynomially…

Amenability and measure of maximal entropy for semigroups of rational maps

- MathematicsGroups, Geometry, and Dynamics
- 2021

In this article we discuss relations between algebraic and dynamical properties of non-cyclic semigroups of rational maps.

Sharing a Measure of Maximal Entropy in Polynomial Semigroups

- Mathematics
- 2020

Let $P_1,P_2,\dots, P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or to Chebyshev polynomials, and $S$ the semigroup under composition…

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