[Paper Review] Combinatorial rigidity for unicritical polynomials

Artur Avila, Jeremy Kahn+2 more|ArXiv.org|Jul 12, 2005

Mathematical Dynamics and Fractals7 references4 citations

TL;DR

This paper establishes combinatorial rigidity for unicritical polynomials $f_c: z \mapsto z^d + c$ that are at most finitely renormalizable and have only repelling periodic points. By proving uniform quasiconformal pseudo-conjugacies between maps with matching combinatorics up to a given scale, and using complex bounds and Teichmüller theory, the authors show that such maps are conformally conjugate, implying local connectivity of the Multibrot set at these parameters—generalizing Yoccoz's theorem to higher-degree polynomials.

ABSTRACT

We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the ``Multibrot set'') is locally connected at the corresponding parameter values. It generalizes Yoccoz's Theorem for quadratics to the higher degree case.

Motivation & Objective

• To establish combinatorial rigidity for unicritical polynomials of degree $d \geq 2$ under finite renormalizability and repelling periodic orbits.
• To generalize Yoccoz’s theorem on local connectivity of the Mandelbrot set to the higher-degree Multibrot sets.
• To show that combinatorially equivalent maps in the unicritical family are conformally equivalent under the given conditions.
• To develop a method based on pseudo-conjugacies and Teichmüller theory that avoids strong geometric control of dynamics.

Proposed method

• Constructs a 'favorite nest' of puzzle pieces adapted to the dynamics of unicritical polynomials.
• Transfers a priori bounds from [KL2] to this favorite nest to control geometry at all scales.
• Uses Teichmüller theory to show that puzzle pieces of combinatorially equivalent maps stay uniformly bounded in Teichmüller distance.
• Establishes uniform quasiconformal pseudo-conjugacies between maps with matching combinatorics up to a given depth.
• Applies a pullback argument to extend pseudo-conjugacies to the full dynamical plane.
• Reduces the general case to the non-renormalizable case via straightening maps in nested little Multibrot copies.

Experimental results

Research questions

• RQ1 Can combinatorial rigidity be established for unicritical polynomials of degree $d \geq 2$ under finite renormalizability and repelling periodic points?
• RQ2 Does the existence of uniformly bounded quasiconformal pseudo-conjugacies imply conformal conjugacy for such maps?
• RQ3 Can the proof technique used in the quadratic case be generalized to higher-degree unicritical polynomials?
• RQ4 Is the Multibrot set locally connected at parameters corresponding to finitely renormalizable unicritical maps with only repelling periodic points?
• RQ5 How does the structure of the parameter space (Multibrot set) relate to combinatorial equivalence classes under these conditions?

Key findings

• Combinatorial rigidity holds for any unicritical polynomial $f_c$ of degree $d \geq 2$ that is at most finitely renormalizable and has only repelling periodic points.
• The Multibrot set $\mathcal{M}_d$ is locally connected at all such parameter values $c$.
• Uniform quasiconformal pseudo-conjugacies exist between maps with matching combinatorics up to a given depth, with dilatation bounds depending only on initial data.
• The proof avoids strong geometric assumptions on the dynamics, relying instead on complex bounds and Teichmüller theory.
• The result generalizes Yoccoz’s theorem on local connectivity of the Mandelbrot set to higher-degree polynomials.
• The method extends to the multicritical case, though the paper focuses on the unicritical setting for clarity.