Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves

TL;DR

This paper proves that the Hausdorff dimension of Julia sets varies real-analytically along parabolic arcs in multicorns and investigates the singularities of algebraic curves related to polynomial dynamics.

Contribution

It establishes the real-analyticity of Hausdorff dimension along parabolic arcs and analyzes the singularities of algebraic curves Per_n(1) in polynomial families.

Findings

Hausdorff dimension is a real-analytic function along parabolic arcs.

Constructed a quasiconformal deformation space for these arcs.

Identified the nature of singularities in algebraic sets Per_n(1).

Abstract

The boundaries of the hyperbolic components of odd period of the multicorns contain real-analytic arcs consisting of quasi-conformally conjugate parabolic parameters. One of the main results of this paper asserts that the Hausdorff dimension of the Julia sets is a real-analytic function of the parameter along these parabolic arcs. This is achieved by constructing a complex one-dimensional quasiconformal deformation space of the parabolic arcs which are contained in the dynamically defined algebraic curves $\mathrm{Per}_n(1)$ of a suitably complexified family of polynomials. As another application of this deformation step, we show that the dynamically natural parametrization of the parabolic arcs has a non-vanishing derivative at all but (possibly) finitely many points. We also look at the algebraic sets $\mathrm{Per}_n(1)$ in various families of polynomials, the nature of their singularities, and the `dynamical' behavior of these singular parameters.