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

摘要

The boundaries of the hyperbolic components of odd period of the multicornscontain real-analytic arcs consisting of quasi-conformally conjugate parabolicparameters. One of the main results of this paper asserts that the Hausdorffdimension of the Julia sets is a real-analytic function of the parameter alongthese parabolic arcs. This is achieved by constructing a complexone-dimensional quasiconformal deformation space of the parabolic arcs whichare contained in the dynamically defined algebraic curves $\mathrm{Per}_n(1)$of a suitably complexified family of polynomials. As another application ofthis deformation step, we show that the dynamically natural parametrization ofthe 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 ofpolynomials, the nature of their singularities, and the `dynamical' behavior ofthese singular parameters.