Polynomial-like semi-conjugates of the shift map

TL;DR

This paper demonstrates that for certain polynomials with Cantor Julia sets, the Julia set can be represented as a minimal quotient of the one-sided shift space, with the projection being as injective as possible.

Contribution

It establishes a semi-conjugacy between polynomial Julia sets and shift spaces, characterizing the Julia set as a minimal quotient with near-injective projection.

Findings

Julia set is a minimal quotient of the shift space

Projection from shift space to Julia set is nearly injective

Provides a new perspective on polynomial dynamics

Abstract

In this paper I prove that for a polynomial of degree $d$ with a Cantor Julia set $J$, the Julia set can be understood as the simplest possible quotiont of the one sided shift space $\Sigma_d$ with dynamics given by the shift. Here simplest possible means that, the projection $\pi: \Sigma_d \Rightarrow J$ is as injective as possible.