arXiv subjects

Willie Rush Lim

Publications and source records attributed to Willie Rush Lim.

7 recordsLinked to original sources

arXiv2026

The combinatorics of sector renormalization

The goal of this note is to systematically develop the fundamental arithmetic and combinatorial properties of the sector renormalization operation on rigid rotations. We employ the specific framework of modified continued fractions appropriate for sector renormalization and analyze their properties. By allowing infinite first return times, this framework yields a dynamical compactification of the space of irrationals called the parabolic compactification; we show that it is characterized by some universal properties. We also discuss the corresponding natural extension and introduce the continuant group, the time semigroup, and topological cascades. For example, we demonstrate how a bi-infinite tower of sector renormalizations of irrational rotations can be packaged within a single dynamical plane as a cascade of translations. This note will serve as a foundational combinatorial tool for studying the geometric properties of sector renormalizations of holomorphic maps with irrationally indifferent fixed points, particularly neutral quadratic polynomials.

math.DS↗

arXiv2026

Rigidity of the attractor of neutral renormalization

We prove combinatorial rigidity for the full renormalization attractor of neutral quadratic polynomials. More precisely, any two bi-infinite renormalization towers with the same combinatorics are conformally conjugate on neighborhoods of their Mother Hedgehogs. The rigidity theorem includes arbitrary irrational combinatorics and respective parabolic enrichments. The proof relies on a comprehensive analysis of neutral cascades, i.e. transcendental dynamical systems arising from the rescaled limits of the first return maps of neutral quadratic polynomials. We establish uniform butterfly bounds for neutral cascades and prove that any two combinatorially equivalent neutral cascades are affinely conjugate. We also prove rigidity for parabolic towers with equivalent backward combinatorics.

math.DS↗

arXiv2026

Lebesgue measure of the postcritical set of neutral quadratic polynomials

We prove that the postcritical set of every quadratic polynomial with a neutral fixed point has zero Lebesgue measure. In particular, no arithmetic condition on the rotation number is required. Our proof uses sector renormalization and the pseudo-Siegel bounds of Dudko-Lyubich to derive uniform distortion estimates for the renormalization change of variables. For sector renormalization towers, we also prove the same zero area statement for the associated postcritical set and the optimal Brjuno criterion for the interior of the associated Mother Hedgehog.

math.DS↗

arXiv2026

Uniform bounds for bubbles of neutral quadratic polynomials

Given a quadratic polynomial with an irrationally indifferent fixed point of bounded type, a bubble is an iterated preimage of its Siegel disk. Similar to pseudo-Siegel disks, one constructs pseudo-bubbles from bubbles by filling in parabolic fjords. We introduce the near-degenerate regime for the $α'$-Dynamics controlling the geometry of these pseudo-bubbles in deep scale. Consequently, we obtain uniform bounds on the size and regularity of pseudo-bubbles independent of the rotation number. In [DLL], jointly with Lyubich, this result serves as a necessary ingredient for establishing uniform butterfly bounds and ultimately the combinatorial rigidity of the full attractor of neutral renormalization.

math.DS↗

arXiv2026

Hyperbolicity of renormalization of critical quasicircle maps

There is a well developed renormalization theory of real analytic critical circle maps by de Faria, de Melo, and Yampolsky. In this paper, we extend Yampolsky's result on hyperbolicity of renormalization periodic points to a larger class of dynamical objects, namely critical quasicircle maps, i.e. analytic self homeomorphisms of a quasicircle with a single critical point. Unlike critical circle maps, the inner and outer criticalities of critical quasicircle maps can be distinct. We develop a compact analytic renormalization operator called Corona Renormalization with a hyperbolic fixed point whose stable manifold has codimension one and consists of critical quasicircle maps of the same criticality and periodic type rotation number. Our proof is an adaptation of Pacman Renormalization Theory for Siegel disks as well as rigidity results on the escaping dynamics of transcendental entire functions.

math.DS↗

arXiv2025

A priori bounds and degeneration of Herman rings with bounded type rotation number

By adapting the near-degenerate regime designed by Kahn, Lyubich, and D. Dudko, we prove that the boundaries of Herman rings with bounded type rotation number and of the simplest configuration are quasicircles with dilatation depending only on the degree and the rotation number. As a consequence, we show that these Herman rings always degenerate to a Herman curve, i.e. an invariant Jordan curve that is not contained in the closure of a rotation domain and on which the map is conjugate to a rigid rotation. This process enables us to construct the first general examples of rational maps having Herman curves of bounded type with arbitrary degree and combinatorics. In particular, they do not come from Blaschke products. We also demonstrate the existence of Renormalization Theory for Herman curves by constructing rescaled limits of the first return maps in the unicritical case.

math.DS↗

arXiv2025

Rigidity of J-rotational rational maps and critical quasicircle maps

We present a number of rigidity results concerning holomorphic dynamical systems admitting rotation quasicircles. Firstly, we show the absence of line fields on the Julia set of any rational map that is geometrically finite away from a number of rotation quasicircles with bounded type rotation number. As an application, we prove combinatorial rigidity associated to the problem of degeneration of Herman rings of the simplest configuration. Secondly, we extend a result of de Faria and de Melo on the $C^{1+α}$ rigidity of critical circle maps with bounded type rotation number to a larger class of dynamical objects, namely critical quasicircle maps. Unlike critical circle maps, critical quasicircle maps may have imbalanced inner and outer criticalities. As a consequence, we prove dynamical universality and exponential convergence of renormalization towards a horseshoe attractor.

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