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 $\Lambda$. The Barnsley--Harrington Mandelbrot set $\mathcal{M}$ is the set of $s\in \mathbb{D}$ for which $\Lambda$ is connected. Let $K$ denote the filled set of a connected $\Lambda$. 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.