Polynomial-like semi-conjugates of the shift map

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.