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May  2017, 37(5): 2565-2588. doi: 10.3934/dcds.2017110

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

 1 Jacobs University Bremen, Campus Ring 1, Bremen 28759, Germany 2 Institute for Mathematical Sciences, Stony Brook University, Stony Brook, 11794, NY, USA

Received  May 2016 Revised  January 2017 Published  February 2017

Fund Project: The author was supported by Deutsche Forschungsgemeinschaft DFG.

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.

Citation: Sabyasachi Mukherjee. Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2565-2588. doi: 10.3934/dcds.2017110
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##### References:
$\mathcal{M}_2^*$, also known as the tricorn and the parabolic arcs on the boundary of the hyperbolic component of period 1 (in blue)
Pictorial representation of the image of $\left[0,1\right]$ under the quasiconformal map $L_w$; for $w=1+i/8$ (top) and $w=1$ (bottom). The Fatou coordinates of $c_0$ and $f_{c_0}^{\circ k} (c_0)$ are $1/4$ and $3/4$ respectively. For $w=1+i/8$, $L_w(1/4)=1/8+i$ and $L_w(3/4)=7/8-i$, and for $w=1$, $L_w(1/4)=1/4+i$ and $L_w(3/4)=3/4-i$. Observe that $L_w$ commutes with $z\mapsto \overline{z}+1/2$ only when $w\in \mathbb{R}$
$\pi_2 \circ F : w \mapsto b(w)$ is injective in a neighborhood of $\widetilde{u}$ for all but possibly finitely many $\widetilde{u} \in \mathbb{R}$
The outer yellow curve indicates part of $\mathrm{Per}_1(1)\cap \lbrace a=\overline{b}\rbrace$, and the inner blue curve (along with the red point) indicates part of the deformation $\mathrm{Per}_1(r)\cap \lbrace a=\overline{b}\rbrace$ for some $r\in (1-\epsilon,1)$. The cusp point $c_0$ on the yellow curve is a critical point of $h_1$, i.e. a singular point of $\mathrm{Per}_1(1)$, and the red point is a critical point of $h_r$; i.e a singular point of $\mathrm{Per}_1(r)$
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