Continuity of core entropy of quadratic polynomials

TL;DR

This paper proves that the core entropy of quadratic polynomials, a combinatorial dynamical invariant, varies continuously with the external angle, enhancing understanding of polynomial parameter spaces.

Contribution

It establishes the continuity of core entropy as a function of external angle for quadratic polynomials, answering a question posed by Thurston.

Findings

Core entropy varies continuously with external angle.

Provides a combinatorial framework for understanding polynomial dynamics.

Extends entropy theory from real to complex quadratic polynomials.

Abstract

The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials. The theory of core entropy extends to complex polynomials the entropy theory for real unimodal maps: the real segment is replaced by an invariant tree, known as Hubbard tree, which lives inside the filled Julia set. We prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle, answering a question of Thurston.