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

Figure 1.  $\mathcal{M}_2^*$, also known as the tricorn and the parabolic arcs on the boundary of the hyperbolic component of period 1 (in blue)

Figure 2.  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 $w\in \mathbb{R}$

Figure 3.  $\pi_2 \circ F : w \mapsto b(w)$ is injective in a neighborhood of $\widetilde{u}$ for all but possibly finitely many $\widetilde{u} \in \mathbb{R}$

Figure 4.  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 $\mathrm{Per}_1(r)$