Pseudo-monodromy and the Mandelbrot set

TL;DR

This paper studies how codings of Julia sets change discontinuously at certain parameters in the Mandelbrot set, linking these changes to kneading sequences and monodromy actions in complex dynamics.

Contribution

It provides a detailed description of the discontinuity of Julia set codings at hyperbolic components, connecting it to kneading sequences and monodromy in the horseshoe locus.

Findings

Discontinuity of codings is characterized in terms of kneading sequences.

Main result describes the behavior at hyperbolic component root points.

Connects Julia set coding discontinuities to monodromy actions in complex dynamics.

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.