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

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 $f_c$.