Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

TL;DR

This paper establishes a relationship between the topological entropy of quadratic polynomials and the Hausdorff dimensions of certain sets of external rays landing on Julia and Mandelbrot set sections, extending to Hubbard trees.

Contribution

It introduces a formula linking entropy and Hausdorff dimensions for real quadratic polynomials and generalizes it to Hubbard trees and specific Mandelbrot set slices.

Findings

Derived a formula connecting entropy and Hausdorff dimension for real quadratic polynomials.

Extended the relationship to external angles landing on slices of the Mandelbrot set.

Provided insights into the structure of Julia sets and Mandelbrot set sections through Hausdorff dimensions.

Abstract

Let c be a real parameter in the Mandelbrot set, and f_c(z):= z^2 + c. We prove a formula relating the topological entropy of f_c to the Hausdorff dimension of the set of rays landing on the real Julia set, and to the Hausdorff dimension of the set of rays landing on the real section of the Mandelbrot set, to the right of the given parameter c. We then generalize the result by looking at the entropy of Hubbard trees: namely, we relate the Hausdorff dimension of the set of external angles which land on a certain slice of a principal vein in the Mandelbrot set to the topological entropy of the quadratic polynomial f_c restricted to its Hubbard tree.