On Yoccoz Return Functions

TL;DR

This paper introduces the Yoccoz return function, a generalized coding tool for the dynamics of complex polynomials with disconnected Julia sets, providing necessary and sufficient conditions for its realization.

Contribution

It generalizes the Yoccoz tau-function to code critical point returns in complex polynomials with disconnected Julia sets, offering a recursive framework.

Findings

Defined the Yoccoz return function for polynomials with disconnected Julia sets.

Established necessary conditions for Yoccoz return functions.

Proved these conditions are sufficient for polynomials with a single bounded critical orbit.

Abstract

We study the dynamics of complex polynomials. We obtain results on Poincare return maps defined on certain neighborhoods of a point with bounded orbit under a polynomial. We introduce a generalization of the Yoccoz tau-function, the Yoccoz return function, which codes the returns of a critical point with bounded orbit of any complex polynomial with a disconnect Julia set. We give necessary conditions on Yoccoz return functions, which allow for the recursive definition of an abstract tau-function. These conditions are also sufficient for polynomials that have a disconnected Julia set and exactly one critical point with bounded orbit.