arXiv subjects

Alex Kapiamba

Publications and source records attributed to Alex Kapiamba.

4 recordsLinked to original sources

arXiv2026

MLC for parabolically bounded primitive renormalization

We prove $\textit{a priori}$ bounds and MLC (local connectivity of the Mandelbrot set $\mathcal{M}$) for a class of infinitely renormalizable parameters whose renormalization type is primitive but can approach the cusp of $\mathcal{M}$. To this end we develop and refine a variety of tools that allow us to control degeneration of renormalizations. They include the Thin-Thick Decomposition, the Value Calculus, the Wanderers Theorem, and the Wave Lemma.

math.DS↗

arXiv2024

The near-parabolic geometry of external rays

Using Lavaurs maps and near-parabolic renormalization, we describe the degenerating geometry of external rays for quadratic polynomials when a periodic cycle becomes parabolic. We similarly describe the geometry of parameter rays for the Mandelbrot set near parabolic points. Using this geometric control we establish new bounds on the size of limbs of the Mandelbrot set, for example providing a quadratic Pommerenke-Levin-Yoccoz inequality in the near-parabolic setting.

math.DS↗

arXiv2022

Non-degenerate near-parabolic renormalization

Invariant classes under parabolic and near-parabolic renormalization have proved extremely useful for studying the dynamics of polynomials. The first such class was introduced by Inou-Shishikura to study quadratic polynomials; their argument has been extended to the unicritical cubic case by Yang and the general unicritical case by Ch\'eritat. However, all of these classes are only applicable to maps which have a fixed point with multiplier close to one, though it is well-known that similar phenomena occur when the multiplier is close to any root of unity. In this paper we define the parabolic and near-parabolic renormalization operators in the general setting and construct invariant classes. In the general setting we can observe a new phenomenon: the multiplier may be close to several roots unity. In this case, we show how to directly relate the different near-parabolic renormalizations that arise.

math.DS↗

arXiv2021

An optimal Yoccoz inequality for near-parabolic quadratic polynomials

Using Lavaurs maps and near-parabolic renormalization, we describe the degenerating geometry of external rays for quadratic polynomials when a periodic cycle becomes parabolic. We similarly describe the geometry of parameter rays for the Mandelbrot set near parabolic points. Using this geometric control we establish new bounds on the size of limbs of the Mandelbrot set, providing a quadratic Pommerenke-Levin-Yoccoz inequality in the near-parabolic setting.

math.DS↗