Parabolic and near-parabolic renormalizations for local degree three

Abstract

The invariant class under parabolic and near-parabolic renormalizations constructed by Inou and Shishikura has been proved to be extremely useful in recent years. It leads to several important progresses on the dynamics of certain holomorphic maps with critical points of local degree two. In this paper, we construct a new class consisting of holomorphic maps with critical points of local degree three which is invariant under parabolic and near-parabolic renormalizations. As potential applications, some results of cubic unicritical polynomials can be obtained similarly as the quadratic case. For example, the existence of cubic unicritical Julia sets with positive area, the characterizations of the topology and geometry of cubic irrationally indifferent attractors etc.

Figures (13)

• Figure 1: Left: The immediate parabolic basin of $P$ contains the critical point $cp_P$ of local degree $3$, where the chessboard structure and some special points are marked. Right: A zoom of the left near the critical point $-\mu$ and the pole $-1$.
• Figure 2: The domains $V$ (the gray part), $V'=U_\eta^P$ with $\eta=3$ and their zooms near $-\mu=4\sqrt{6}-11{\,(\doteqdot{-1.2020}\ldots)}$ and $-1$. The outer boundary of $V'$ looks like a circle. The half widths of these pictures are $450$, $0.08$, $10^{-3}$ and $10^{-4}$ respectively.
• Figure 3: Riemann surface $X$ (left) and the domain $Y$ (right).
• Figure 4: The commutative diagram about the maps $Q$, $\varphi$ and their lifts $\widetilde{Q}$, $\widetilde{\varphi}$.
• Figure 5: Various domains for $F=Q$ (i.e., $\varphi=id$). The images of $D_1$ and $D_1^+$ under $g$ (i.e., the inverse of $F$) are denoted by $D_{-n}=g^n(D_0)$, $D_{-n}'=g^n(D_0')$, $D_{-n}"=g^n(D_0")$, $D_{-n}"'=g^{n-1}(D_{-1}"')$, $D_{-n}""=g^{n-1}(D_{-1}"")$, $D_{-n}^+=g^n(D_0^+)$ and their projection by $\pi_X$ are drawn. To emphasize details, we choose $\eta=1.5$ in this figure.

Theorems & Definitions (72)

• Definition : The class $\mathcal{F}_1$
• Corollary
• Theorem 2.1: and the definitions of Fatou coordinates and horn maps, Shi00a
• Definition : Parabolic renormalization
• Proposition 2.2: see Figure \ref{['Fig-P:chessboard']}
• proof
• Definition : Some intermediate mappings and the mapping $Q$
• Definition : $V'=U_\eta^P$ and $U_\eta^Q$, see Figure \ref{['Fig-V']}
• Definition : Disk $\mathscr{D}$ and domain $V$
• Proposition 3.1: Relation between $V$ and $V'$