Hausdorff Dimension of Julia Sets in the Logistic Family

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 \({{\mathcal {M}}}\), 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 \({{\mathcal {M}}}\) for most real parameters in the the sense of 1-dimensional Lebesgue measure.