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

Jun Luo, Bo Tan, Yi Yang, Xiao-Ting Yao

Published 2023-01-30Version 1

The classical core entropy for post critically finite (PCF) polynomials f with degree no less than two is defined to be the topological entropy of f restricted to its Hubbard tree. We fully generalize this notion by a new quantity, called the (general) core entropy, which is well defined whenever f has a connected Julia set. If f is PCF, the core entropy equals the classical version. If two polynomials f and g are J-equivalent then they share the same core entropy. If f is unicritical and has no irrationally neutral cycle, we can identify a compact subset of the unit circle invariant under the doubling map whose Hausdorff dimension equals the ratio of the core entropy to log(d). We also carefully analyze the function that sends every parameter c in the Mandelbrot set to the core entropy of the polynomial z^2+c.