Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $Per_n(1)$ curves

Pages: 2565 - 2588, Volume 37, Issue 5, May 2017      doi:10.3934/dcds.2017110

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Sabyasachi Mukherjee - Jacobs University Bremen, Campus Ring 1, Bremen 28759, Germany (email)

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.

Keywords:  Hausdorff dimension, parabolic curves, antiholomorphic dynamics, quasiconformal deformation, multicorns.
Mathematics Subject Classification:  Primary: 37F10, 37F30, 37F35, 37F45.

Received: May 2016;      Revised: January 2017;      Available Online: February 2017.