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Wandering continua, renormalization, and Lebesgue measure on Julia sets
Willie Rush Lim
We study the Lebesgue measure of wandering continua and related invariant sets in Julia sets, using the hyperbolic area method recently developed by Ye. For entire functions of fin…
Rigidity of the attractor of neutral renormalization
Dzmitry Dudko, Willie Rush Lim, Mikhail Lyubich
We prove combinatorial rigidity for the full renormalization attractor of neutral quadratic polynomials. More precisely, any two bi-infinite renormalization towers with the same co…
Lebesgue measure of the postcritical set of neutral quadratic polynomials
Willie Rush Lim
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…
Uniform bounds for bubbles of neutral quadratic polynomials
Dzmitry Dudko, Willie Rush Lim
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…
The combinatorics of sector renormalization
Willie Rush Lim
The goal of this paper is to systematically develop the fundamental arithmetic and combinatorial properties of the sector renormalization operation on rigid rotations. We employ th…