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

Abstract

For all polynomials $f$ with $\mathrm{deg}(f)\ge 2$ that have a connected filled Julia set $K$, we introduce a new quantity $h_{\mathrm{GCE}}(f)$, such that $h_{\mathrm{GCE}}\left(f^n\right)=n\cdot h_{\mathrm{GCE}}(f)$ for all $n\ge 1$ and $h_{\mathrm{GCE}}(f)=h_{\mathrm{GCE}}(g)$ for $J$-equivalent $f$ and $g$. When the coefficients and the critical points of $f$ are real, $h_{\mathrm{GCE}}(f)=h(K\cap \mathbb{R},f)$. When $f$ is post-critically finite, $h_{\mathrm{GCE}}(f)$ 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 $c\mapsto h_{\mathrm{GCE}}(f_c)$ is not continuous. However, its lower envelope $h_{\mathrm{core}}:\mathcal{M}\to \mathbb{R}$ given by $h_{\mathrm{core}}(c)=\inf \left\{t:\exists c_n\ne c\text{with}{c}_n\to c\text{and}t=\underset{n\to \infty }{\lim }{h}_{\mathrm{GCE}}\left(f_{c_n}\right)\right\}$ is continuous over $\mathcal{M}$ and has three properties. First, every $h_{\mathrm{core}}^{-1}([0,s])$ with $s\ge 0$ is connected. In particular, $h_{\mathrm{core}}^{-1}(0)$ coincides with the central molecule. Second, $h_{\mathrm{core}}(c)=h(\mathbb{R},f_c)$ for $c\in [-2,\frac{1}{4}]$. Third, $h_{\mathrm{core}}(c)=h(\mathcal{H}(f_c),f_c)$ for post-critically finite $f_c$.

40 pages, 1 figure