Hausdorff dimension of Julia sets in the logistic family

TL;DR

This paper investigates the Hausdorff dimension of Julia sets within the logistic family, revealing continuous dependence on parameters and providing explicit bounds near the Mandelbrot set's tip.

Contribution

It demonstrates the continuous dependence of quadratic Julia set dimensions on parameters and offers explicit bounds at the Mandelbrot set's tip for most real parameters.

Findings

Hausdorff dimension depends continuously on parameter c

Explicit bounds for dimensions near the Mandelbrot tip

Discontinuity of dimension at the Mandelbrot tip is generally avoided

Abstract

A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set M, the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on $c$ and find explicit bounds at the tip of M for most real parameters in the the sense of 1-dimensional Lebesgue measure.