Polynomial semiconjugacies, decompositions of iterations, and invariant curves

TL;DR

This paper investigates polynomial functional equations, characterizes solutions using Julia sets, and applies findings to invariant curves and polynomial iteration decompositions.

Contribution

It provides a new description of solutions to polynomial semiconjugacy equations using Julia sets and extends results on invariant curves and polynomial iteration decompositions.

Findings

Solutions characterized via filled-in Julia sets of B

Structural results on solutions to polynomial semiconjugacies

Applications to invariant curves and polynomial iteration decompositions

Abstract

We study the functional equation $A\circ X=X\circ B$, where $A,$ $B$, and $X$ are polynomials over $\mathbb C$. Using previous results of the author about polynomials sharing preimages of compact sets, we show that for given $B$ its solutions may be described in terms of the filled-in Julia set of $B$. On this base, we prove a number of results describing a general structure of solutions. The results obtained imply in particular the result of Medvedev and Scanlon about invariant curves of maps $F:\,\mathbb C^2 \rightarrow \mathbb C^2$ of the form $(x,y)\rightarrow (f(x),f(y))$, where $f$ is a polynomial, and a version of the result of Zieve and M\"uller about decompositions of iterations of a polynomial.