To Define the Core Entropy for All Polynomials Having a Connected Julia Set

TL;DR

This paper introduces a new entropy measure for polynomials with connected Julia sets, explores its properties, and examines its behavior across the Mandelbrot set, revealing continuity and structural insights.

Contribution

It defines a generalized core entropy for all such polynomials, analyzes its properties, and studies the entropy map's continuity over the Mandelbrot set.

Findings

The entropy map is discontinuous, but its lower envelope is continuous.

The lower envelope's level sets are connected.

The lower envelope matches real and Hubbard tree entropies in specific cases.

Abstract

For all polynomials $f$ with ${\rm deg}(f)\ge2$ that have a connected filled Julia set $K$, we introduce a new quantity $h_{\rm GCE}(f)$, such that $h_{\rm GCE}\left(f^n\right)=n\cdot h_{\rm GCE}(f)$ for all $n\ge1$ and $h_{\rm GCE}(f)=h_{\rm GCE}(g)$ for $J$-equivalent $f$ and $g$. When the coefficients and the critical points of $f$ are real, $h_{\rm GCE}(f)=h(K\cap\mathbb{R},f)$. When $f$ is post-critically finite, $h_{\rm GCE}(f)$ equals the core entropy $h(\mathcal{H}(f),f)$, where $\mathcal{H}(f)$ is the Hubbard tree. For $f_c(z)=z^2+c$ with $c$ varying in the Mandelbrot set $\mathcal{M}$, the entropy map $c\mapsto h_{\rm GCE}(f_c)$ is not continuous. However, its lower envelope $h_{\rm core}:\mathcal{M}\rightarrow\mathbb{R}$ given by $h_{\rm core}(c)=\inf\left\{t:\ \exists\ c_n\ne c\ \text{with}\ c_n\rightarrow c\ \text{and}\ t=\lim\limits_{n\rightarrow\infty}h_{\rm GCE}\left(f_{c_n}\right)\right\}$ is continuous over $\mathcal{M}$ and has three properties. First, every $h_{\rm core}^{-1}([0,s])$ with $s\ge0$ is connected. In particular, $h_{\rm core}^{-1}(0)$ coincides with the central molecule. Second, $h_{\rm core}(c)=h(\mathbb{R},f_c)$ for $c\in[-2,\frac14]$. Third, $h_{\rm core}(c)=h(\mathcal{H}(f_c),f_c)$ for post-critically finite $f_c$.