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 Pern(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 Pern(1) in various families of polynomials, the nature of their singularities, and the 'dynamical' behavior of these singular parameters.1991 Mathematics Subject Classification. Primary: 37F10, 37F30, 37F35, 37F45.

Discontinuity of Straightening in Anti-holomorphic Dynamics: I

“…Moreover, the Fatou vector can be quasi-conformally deformed giving rise to an analytic family of q.c. equivalent parabolic maps [Muk15c,§3]. Heuristically speaking, if straightening maps were continuous, they would preserve the geometry of the parameter space.…”