Some invariant classes for parabolic renormalization in the multicritical case

TL;DR

The paper extends invariant-class theory for parabolic renormalization from the unicritical, unit-disk Blaschke setting to multicritical maps conjugate on their immediate basins to finite Blaschke products. It introduces Dynτ, composition types, descendants, and gleanings to organize how critical points are distributed under renormalization and proves that, for f in Dynτ, the renormalized map Rf lands in Dynτ' for some descendant τ' of τ (provided Rf is not exceptional). The work develops two proofs (via finite-type maps and Blaschke-model connections) to show Dom(Rf) is simply connected and Rf is a finite-type branched covering with finitely many critical values, enabling a precise description of how Rf acts on the sequence of virtual basins and yields a disjoint collection of descendant types on the renormalized basins. A generalisation to nonzero rotation numbers and multiple axis cycles is outlined, establishing a framework for iterated renormalization and the structure of nested pearl necklaces across slices of parameter spaces. Overall, this starts a rigorous program to understand perturbations and perturbative limits (parabolic enrichment) for larger, multicritical families beyond Shishikura’s Dyn2, using Blaschke-models and horn-map equivalence as the core tools.

Abstract

Parabolic renormalization associates to a holomorphic map f with a parabolic fixed point another holomorphic map with a parabolic fixed point. This procedure is essential for understanding the phenomenon of parabolic enrichment, which occurs when one perturbs f appropriately. Shishikura defined in [Shi98] (see also [LY14]) a class of maps that is stable under this parabolic renormalization operator. These maps have only one critical point in their immediate basins. We extend here this result to the more general classes of maps conjugated on their immediate basins to finite Blaschke products. For these classes, we introduce an analogue of Milnor's mapping schemes [Mil12] and describe the action of parabolic renormalization on the scheme. This shall be a starting point to the fine study of perturbation of these bigger classes of maps.

Figures (14)

• Figure 1: Slice $\mathrm{Per}_1(e^{2\pi i 2/5})$ of the cubic polynomials. We took Zakeri's parametrization Zakeri: $P_c = \lambda z(1-\frac{1+c}{2c}z+\frac{1}{3c}z^2)$, for which $0$ is fixed of multiplier $\lambda$ and the set of finite critical points is $\{1,c\}$. In blue the bitransitive case, in amber the adjacent case, in champagne the capture case and in different shades of white and grey the disjoint case: white when one critical point escapes to infinity, dark grey for the bifurcation locus, light grey for the rest. A tiny black dot marks the parameter $c=0$, which does not belong to the parameter space $\mathbf{C}^*$ of the slice. The 5 exceptional parameters are the contact points of the pearls (blue or amber components). A deeper description of the slices $\mathrm{Per}_1(e^{2\pi i p/q})$ can be found in runze3.
• Figure 2: $\mathrm{Per}_1(1)$ in log-coordinates, an enrichment and its infinite pearl necklace.
• Figure 3: Schematic representation of the pearl necklaces associated to successive enrichments in the parameter plane. The top row represents the pearl necklace of the family $c\mapsto P_c$, in the space $\mathrm{Per}_1(\exp(2\pi i 1/3))$, drawn in the $\log(c)/i$ coordinate, i.e. as on Figure \ref{['fig:per1implo']}. The second row shows the pearl necklace of the family $c\mapsto R_{0/1} P_c$. It is necessarily contained in the pearls of the family $c\mapsto P_c$. In this simplified diagram we omit the contact points between the regions and their sub-region, though are an interesting aspect of the process. The third row shows the pearl necklace of the family $c\mapsto R_{0/1} R_{0/1} P_c$. The amber pearls represent the maps that are of composition type $3$ while the blue ones of composition type $2\circ 2$. Type E parameters (two immediate basin components) are at the contact points between pearls. The rest is of type $2$ (iff. there is only one critical point in the immediate basin).
• Figure 4: A non-realistic sketch of the lift $U$ by $E:z\mapsto e^{2\pi iz}$ of the immediate basin of some $Rf$, such that $U$ would have a real part that is not bounded from below, nor above.
• Figure 5: $(1,3,4,8)$ as a gleaning of $(10,7)$. The black dots are the pebbles, the bowls are draw in solid lines and the purses in dashed lines.

Theorems & Definitions (113)

• Theorem 1.1: Shishikura
• Definition 1.2
• Definition 1.3
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Proposition 2.4
• Remark 2.5
• Lemma 2.6