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

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.

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References in corpus (5)

• On Multicorns and Unicorns II: Bifurcations in Spaces of Antiholomorphic Polynomials
• Non-landing parameter rays of the multicorns
• Green's function and anti-holomorphic dynamics on a torus
• On a Family of Rational Perturbations of the Doubling Map
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Cited by in corpus (2)

• Schwarz reflections and the Tricorn
• Discontinuity of Straightening in Anti-holomorphic Dynamics: I