Laminations and External Angles for Similarity Pairs

Abstract

A {\em similarity pair} is the dynamical system in $\mathbb{C}$ generated by two maps $f:z \to sz-1$ and $g:z \to sz+1$ for $|s|<1$. Associated to the dynamical system is an attractor $Λ$. The Barnsley--Harrington Mandelbrot set $\mathcal{M}$ is the set of $s\in \mathbb{D}$ for which $Λ$ is connected. Let $K$ denote the filled set of a connected $Λ$. For $s\in \partial \mathcal{M}$ we show that the (partially defined) action of the semigroup on $\partial K$ is topologically conjugate to a (discontinuous) piecewise linear action of constant slope. Conditional on a conjecture (satisfied for `most' $s\in \partial \mathcal{M}$) we give a necessary and sufficient condition in terms of the dynamics on $\partial K$ for $K$ to contain cut points, and we describe the set of all such cut points in terms of infinite walks in a directed graph $\textrm{IG}$ obtained by an explicit recursive algorithm. The structure of the `dynamical cut point set' for a 2-dimensional family of piecewise linear actions (containing those coming from $s\in \partial \mathcal{M}$) recovers and generalizes the Douady--Hubbard--Thurston quadratic minor lamination for the abstract Mandelbrot set.

Disclosure

“6.3. 1.3. Acknowledgements. We would like to thank Bernat Espigulé, Toby Hall, Curt Mc- Mullen, Steffen Rohde, Victor Sirvent and Boris Solomyak for valuable comments, assis- tance and encouragement. We would also like to acknowledge that Claude was used to write and/or modify code used for the numerical investigations carried out in this paper, and also to create some of the figures that appear in the Appendix. 2. Definitions Let D ⊂ C de”

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Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-23T05:12:48.476353Z

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