[Paper Review] Hyperbolic components of McMullen maps

Weiyuan Qiu, Pascale Roesch+2 more|arXiv (Cornell University)|Jul 2, 2012

Mathematical Dynamics and Fractals19 references3 citations

TL;DR

This paper proves that the boundaries of all hyperbolic components in the parameter space of McMullen maps $ f_\lambda(z) = z^n + \lambda z^{-n} $ (for $ n \geq 3 $) are Jordan curves, resolving a long-standing problem posed by Devaney. As a consequence, cusps—parameters with parabolic cycles on the boundary of the immediate attracting basin of infinity—are shown to be dense on the boundary of the unbounded hyperbolic component, providing a dynamical analogue to McMullen’s theorem on Teichmüller space boundaries.

ABSTRACT

In this article, we study the hyperbolic components of McMullen maps. We show that the boundaries of all hyperbolic components are Jordan curves. This settles a problem posed by Devaney. As a consequence, we show that cusps are dense on the boundary of the unbounded hyperbolic component. This is a dynamical analogue of McMullen's theorem that cusps are dense on the Bers' boundary of Teichmüller space.

Motivation & Objective

• To resolve a problem posed by Devaney regarding the topological structure of hyperbolic component boundaries in McMullen maps.
• To establish that all hyperbolic components in the parameter space of $ f_\lambda(z) = z^n + \lambda z^{-n} $ have Jordan curve boundaries.
• To demonstrate that cusps are dense on the boundary of the unbounded hyperbolic component $ \mathcal{H}_0 $, where the Julia set is a Cantor set.
• To provide a dynamical analogue of McMullen’s theorem on the density of cusps on the Bers’ boundary of Teichmüller space.
• To develop a new approach using dynamical Yoccoz puzzles and continuous motion of cut rays to prove local connectivity and Jordan curve structure.

Proposed method

• Utilizes the Yoccoz puzzle structure induced by 'cut rays'—Jordan curves constructed by Devaney—that vary continuously in the Hausdorff topology with respect to the parameter $ \lambda $.
• Employs quasiconformal conjugacies between maps in the same impression of a parameter ray, constructed via cut rays, to prove local connectivity of the boundary.
• Applies a 'zero measure argument' based on extremal length and the area-modulus inequality to show that the Julia set minus preimages of a certain repelling system has zero Lebesgue measure.
• Defines a repelling system $ g: \mathbf{U} \to \mathbf{V} $ on a nested sequence of domains $ \mathbf{V}^d = g^{-d}(\mathbf{V}) $, with $ \mathbf{V}^{d+1} \Subset \mathbf{V}^d $, to analyze the structure of the Julia set and critical orbits.
• Uses the modulus $ \mathbf{m}(A) $ of annuli $ \mathbf{A}^d(z) = \mathbf{V}^d(z) \setminus \overline{\mathbf{V}^{d+1}} $ to show $ \sum_{d \geq 1} \mathbf{m}(\mathbf{A}^d(z)) = \infty $, implying the Julia set is a Cantor set of zero measure.
• Applies the area-modulus inequality $ \text{area}(\mathbf{V}^d) \leq \frac{\text{area}(\mathbf{V})}{1 + 4\pi M_d} $, where $ M_d \to \infty $, to conclude $ \text{area}(\mathbf{V}^d) \to 0 $, supporting the zero measure result.

Experimental results

Research questions

• RQ1 Are the boundaries of all hyperbolic components in the parameter space of McMullen maps topologically Jordan curves?
• RQ2 Is the set of cusp parameters dense on the boundary of the unbounded hyperbolic component $ \mathcal{H}_0 $, where the Julia set is a Cantor set?
• RQ3 Can the boundary of the unbounded hyperbolic component be characterized in terms of dynamical properties such as the presence of parabolic cycles or the free critical orbit?
• RQ4 Does the dynamical Yoccoz puzzle structure, combined with continuous motion of cut rays, suffice to prove local connectivity and Jordan curve structure of hyperbolic component boundaries?
• RQ5 Is the Julia set of the associated repelling system $ g $ a Cantor set of zero Lebesgue measure, implying the full Julia set is a Cantor set?

Key findings

• The boundaries of all hyperbolic components of the McMullen map $ f_\lambda(z) = z^n + \lambda z^{-n} $ for $ n \geq 3 $ are proven to be Jordan curves, affirming a conjecture by Devaney.
• The boundary $ \partial\mathcal{H}_0 $ of the unbounded hyperbolic component is a Jordan curve, and cusps are dense on it, as shown via the characterization that $ \nu(\theta) $ is a cusp iff $ n^p\theta \equiv \theta \mod \mathbb{Z} $ for some $ p \geq 1 $.
• A parameterization $ \nu: \mathbb{S} \to \partial\mathcal{H}_0 $ is constructed via landing points of external rays, and the set of $ \theta $ satisfying $ n^p\theta \equiv \theta \mod \mathbb{Z} $ is dense in $ \mathbb{S} $, implying dense cusps.
• The Julia set $ J(f_\lambda) $ is a Cantor set of zero Lebesgue measure when $ \lambda \in \partial\mathcal{H}_0 $, as established through a zero measure argument using modulus and area inequalities.
• The repelling system $ g: \mathbf{U} \to \mathbf{V} $ defined on nested domains $ \mathbf{V}^d $ satisfies $ \sum_{d \geq 1} \mathbf{m}(\mathbf{A}^d(z)) = \infty $, implying $ K(g) $ is a Cantor set.
• The area-modulus inequality implies $ \text{area}(\mathbf{V}^d) \to 0 $ as $ d \to \infty $, confirming that the set $ K(g) $ has zero Lebesgue measure, which is key to the main result.