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arXiv · 2405.02204

Pseudo-monodromy and the Mandelbrot set

Yutaka Ishii·Thomas Richards
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Abstract

We investigate the discontinuity of codings for the Julia set of a quadratic map. To each parameter ray, we associate a natural coding for Julia sets on the ray. Given a hyperbolic component $H$ of the Mandelbrot set, we consider the codings along the two parameter rays landing on the root point of $H$. Our main result describes the discontinuity of these two codings in terms of the kneading sequences of the hyperbolic components which are conspicuous to $H$. This result can be interpreted as a solution to the degenerate case of the monodromy action conjecture in the horseshoe locus for the complex H\'enon family.

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Yutaka Ishii, Thomas Richards. 2024-05-03. Pseudo-monodromy and the Mandelbrot set. https://arxiv.org/abs/2405.02204

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