Real-analyticity of Hausdorff Dimension of Julia Sets along the Parabolic Arcs of the Multicorns

Sabyasachi Mukherjee

Published 2014-10-05Version 1

In this article, we take a dimension-theoretic look at the connectedness loci of unicritical anti-polynomials, known as the multicorns and prove a regularity property of the Hausdorff dimension of the Julia sets on the persistently parabolic part of these parameter spaces. The boundaries of the odd period hyperbolic components of the multicorns contain real-analytic arcs consisting of quasi-conformally conjugate parabolic parameters. The principal result of this paper asserts that the Hausdorff dimension of the Julia sets is a real-analytic function of the parameter along these parabolic arcs. We also prove, along the way, that the dynamically natural parametrization of the parabolic arcs has a non-vanishing derivative at all but (possibly) finitely many points. Our main result remains true for more general parabolic loci provided all the active critical points converge to attracting or parabolic cycles.