On the Combinatorial Rigidity for Polynomials with Attracting Cycles

TL;DR

This paper investigates the conditions under which polynomials with attracting cycles are combinatorially rigid, revealing that certain hyperbolic polynomials are not rigid unless they are of a specific disjoint type.

Contribution

It establishes new criteria for combinatorial rigidity in polynomials with attracting cycles, especially characterizing hyperbolic polynomials of connected Julia sets.

Findings

Polynomials with attracting cycles attracting multiple critical points are not rigid.

Hyperbolic polynomials with connected Julia sets are rigid iff of disjoint type.

Provides a characterization of rigidity based on cycle and critical point dynamics.

Abstract

We show that every polynomial of degree $d \geq 2$ in the connectedness locus with an attracting cycle which attracts at least two critical points and no indifferent cycles is not combinatorially rigid. In particular, we prove that a hyperbolic polynomial with connected Julia set is combinatorially rigid if and only if it is of the ``disjoint type''.