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Peter Makienko

Publications and source records attributed to Peter Makienko.

13 recordsLinked to original sources
arXiv2023

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

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.

math.DS↗
arXiv2020

Elementary characters on semigroups: the rational case

Since polynomials form a subsemigroup of the semigroup of rational functions, every character on rational functions is a character on polynomials. On the other direction, not every character on polynomials is the restriction of a character on rational functions. What are the characters on polynomials that can be extended to rational functions? In this work, we conjecture that the only characters that can be extended are those that depends on the degree, often called elementary. Also, we construct two example of character on polynomials, not elementaries, that cannot be extended to rational functions.

math.RA↗
arXiv2019

On hyperbolic cobordisms and Hurwitz classes of holomorphic coverings

In this article we show that for every collection $\mathcal{C}$ of an even number of polynomials, all of the same degree $d>2$ and in general position, there exist two hyperbolic $3$-orbifolds $M_1$ and $M_2$ with a Möbius morphism $α:M_1\rightarrow M_2$ such that the restriction of $α$ to the boundaries $\partial M_1$ and $\partial M_2$ forms a collection of maps $Q$ in the same conformal Hurwitz class of the initial collection $\mathcal{C}$. Also, we discuss the relationship between conformal Hurwitz classes of rational maps and classes of continuous isomorphisms of sandwich products on the set of rational maps.

math.DS↗
arXiv2019

On the fixed points of the Ruelle operator

We discuss the relation between the existence of fixed points of the Ruelle operator acting on different Banach spaces, with Sullivan's conjecture in holomorphic dynamics.

math.DS↗
arXiv2015

On hyperbolic metric and asymptotically finite invariant differentials in holomorphic dynamics

Given a rational map $R$, we consider the complement of the postcritical set $S_R$. In this paper we discuss the existence of invariant Beltrami differentials supported on a $R$ invariant subset $A$ of $S_R$. Under some geometrical restrictions, either on the hyperbolic geometry of $A$ or on the asymptotic behavior of infinitesimal geodesics of the Teichmüller space of $S_R$, we show the absence of invariant Beltrami differentials supported on $A$. In particular, we show that if $A$ has finite hyperbolic area, then $A$ can not support invariant Beltrami differentials except in the case where $R$ is a Lattès map.

math.DS↗
arXiv2013

On Poincaré extensions of rational maps

There is a classical extension, of Möbius automorphisms of the Riemann sphere into isometries of the hyperbolic space $\mathbb{H}^3$, which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of $\mathbb{H}^3$ exploiting the fact that any holomorphic covering between Riemann surfaces is Möbius for a suitable choice of coordinates. We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a product in $\mathbb{H}^3$ that allows to construct a visual extension of any given rational map.

math.DS↗
arXiv2011

On decomposable rational maps

If $R$ is a rational map, the Main Result is a uniformization Theorem for the space of decompositions of the iterates of $R$. Secondly, we show that Fatou conjecture holds for decomposable rational maps.

math.DS↗
arXiv2010

Semigroup representations in holomorphic dynamics

We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.

math.DS↗
arXiv2009

On dynamical Teichmuller spaces

Following ideas from a preprint of the second author, see [2], we investigate relations of dynamical Teichmuller spaces with dynamical objects. We also establish some connections with the theory of deformations of inverse limits and laminations in holomorphic dynamics, see [1]

math.DS↗
arXiv2005

Poincare Series and instability of exponential maps

We relate the properties of the postsingular set for the exponential family to the questions of stability. We calculate the action of the Ruelle operator for the exponential family. We prove that if the asymptotic value is a summable point and its orbit satisfies certain topological conditions, the map is unstable hence there are no Beltrami differentials in the Julia set. Also we show that if the postsingular set is a compact set, then the singular value is summable.

math.DS↗