On the Hausdorff dimension of Julia sets of some real polynomials

TL;DR

This paper investigates the Hausdorff dimension of Julia sets for certain real polynomials, establishing a lower bound for the supremum of these dimensions across real parameters.

Contribution

It proves that for even degree polynomials, the maximum Hausdorff dimension of Julia sets exceeds a specific bound, advancing understanding of fractal geometry in complex dynamics.

Findings

Supremum of Hausdorff dimension exceeds 2d/(d+1) for real c

Provides bounds for Julia set dimensions of real polynomials

Enhances knowledge of fractal complexity in polynomial dynamics

Abstract

We show that the supremum for $c$ real of the Hausdorff dimension of the Julia set of the polynomial $z\mapsto z^d+c$ ($d$ is an even natural number) is greater than $2d/(d+1)$.