Lower bounds on the Hausdorff dimension of some Julia sets

TL;DR

This paper introduces an algorithm for rigorously computing lower bounds on the Hausdorff dimensions of Julia sets for various holomorphic maps, applying it to specific quadratic polynomials and verifying a conjecture about the smoothness of the dimension function.

Contribution

The paper develops a new algorithm for lower bounds on Julia set dimensions and applies it to key quadratic polynomials, including the Feigenbaum polynomial, also confirming a conjecture on the smoothness of the dimension function.

Findings

Lower bounds for Julia set dimensions of certain quadratic polynomials obtained.

Constructed a piecewise constant function providing bounds for all quadratic polynomials in [-2,2].

Confirmed the conjecture that the Hausdorff dimension varies smoothly with the parameter c in a specific interval.

Abstract

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial $p_{\,\mathrm{Feig}}(z)=z^2+c_{\,\mathrm{Feig}}$. In addition to that, we construct a piecewise constant function on $[-2,2]$ that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials $p_c(z) = z^2+c$ with $c \in [-2,2]$. Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial $p_c(z)=z^2+c$, is a $C^1$-smooth function of the real parameter $c$ on the interval $c\in(c_{\,\mathrm{Feig}},-3/4)$.