[Paper Review] Bounded hyperbolic components of bicritical rational maps

Hongming Nie, Kevin M. Pilgrim|arXiv (Cornell University)|Mar 21, 2019

Mathematical Dynamics and Fractals14 references4 citations

TL;DR

This paper proves that hyperbolic components of bicritical rational maps—those with two distinct attracting cycles, each of period at least two—are bounded in the moduli space. Using arithmetic methods in non-Archimedean dynamics, particularly Berkovich space dynamics and the refined Fatou-Shishikura inequality, the authors derive a contradiction from assuming unboundedness, thereby establishing compact closure of such components.

ABSTRACT

We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.

Motivation & Objective

• To determine whether hyperbolic components of bicritical rational maps with two distinct attracting cycles of period ≥2 are bounded in the moduli space.
• To extend Epstein’s boundedness result for quadratic strict type D components to higher-degree bicritical maps.
• To establish compact closure of such hyperbolic components using arithmetic and non-Archimedean dynamical methods.
• To analyze degenerating sequences via holomorphic families and rescaling limits in Berkovich space.

Proposed method

• Utilizes a holomorphic one-parameter family $ f_t $ with $ t o 0 $ to model degeneration, leveraging semi-algebraic structure of the hyperbolic component.
• Applies Berkovich space dynamics to study asymptotics of $ f_t $ as $ t o 0 $, identifying rescaling limits via type II repelling cycles.
• Employs Rivera-Letelier’s arithmetic results on critical orbit behavior in non-Archimedean dynamics to constrain the dynamics of the limit map $ g $.
• Uses the refined Fatou-Shishikura inequality to rule out multiple nonrepelling cycles in unicritical polynomials.
• Analyzes the structure of periodic Fatou components and their boundaries in the Berkovich dynamical system to derive contradictions under unboundedness assumptions.
• Applies case analysis on the limit cycles $ ilde{z} $ and $ ilde{w} $, distinguishing cases based on their intersection with the Julia set and the boundary of the unique fixed Rivera domain.

Experimental results

Research questions

• RQ1 Under what conditions are hyperbolic components of bicritical rational maps bounded in the moduli space?
• RQ2 Can Epstein’s analytic approach to boundedness in the quadratic case be extended to higher-degree bicritical maps using arithmetic methods?
• RQ3 What constraints do two bounded multipliers of distinct attracting cycles impose on the dynamics of a rescaling limit in non-Archimedean dynamics?
• RQ4 How do the dynamics of the rescaling limit $ g $, derived from a degenerating holomorphic family, constrain the critical point behavior in the limit?
• RQ5 Is it possible for a bicritical map to have two distinct non-fixed attracting cycles and still lie in an unbounded hyperbolic component?

Key findings

• The hyperbolic component $ ewcommand{H}{\mathcal{H}} \mathcal{H} \subset \mathcal{M}_d $ of bicritical rational maps with two distinct attracting cycles of period at least two has compact closure in $ \mathcal{M}_d $.
• The existence of two bounded multipliers for distinct cycles forces the rescaling limit $ g $ to satisfy an overdetermined system of dynamical constraints on its critical points.
• In all cases of degeneration, the dynamics of the limit map $ g $ lead to a contradiction with the refined Fatou-Shishikura inequality, which allows at most one nonrepelling cycle in a unicritical polynomial.
• The analysis shows that two distinct nonrepelling cycles in the limit cannot coexist without violating the uniqueness of the nonrepelling cycle in a unicritical polynomial.
• The case where both cycles limit to the same parabolic point leads to a contradiction because such a point has multiplicity three, violating the count in the refined FSI.
• The proof establishes that $ \mathcal{H} $ is bounded by showing that any degenerating sequence leads to a contradiction via the interplay of Berkovich dynamics, arithmetic constraints, and the refined Fatou-Shishikura inequality.