## Notes of Danny Calegari’s Cambridge lecture 12-05-2017

Laminations and external angles for similarity pairs

Joint with Alden Walker.

Ending laminations have an analogue inholomorphic dynamics, impression laminations (aka Thurston lamination).

Similarity pairs is a theory intermediate between holomorphic dynamics and Kleinian groups. I explain the analogue of an ending lamination in this theory.

Example: uniformizing the complement of a locally connected Julia set gives rise to a surjective map of the circle to its boundary. Connecting by geodesics in hyperbolic planes tuples of points mapped to the same point gives rise to a lamination of hyperbolic plane.

1. Similarity pairs

Example of a similarity pair. Two affine maps of ${{\mathbb C}}$ with the same stretch ${c\in{\mathbb C}}$, ${|c|<1}$, but distinct fixed points. E.g. ${f(z)=cz+1}$ and ${g(z)=cz-1}$. Let ${G}$ be the semi-group they generate. It has a limit set ${\Lambda}$, a compact set which is forward invariant under ${G}$.

Easy to draw: start with a large disk, take its image by the ${n}$-ball of ${G}$ (a union of little disks), for ${n}$ large (indeed, ${\Lambda}$ is the intersection of these as ${n}$ tends to infinity).

1.1. Connectivity

The Barnsley-Harrington Mandelbrot set is the set of parameters ${c}$ for which the ${\Lambda}$ is connected. Equivalently if ${f\Lambda\cap g\Lambda\not=\emptyset}$. Equivalently, for which ${c}$ is a root of a power series with coefficients in ${\{-1,0,1\}}$. Equivalently, for which ${c}$ is a limit of roots of polynomials with coefficients in ${\{-1,0,1\}}$.

How do roots arise ? Start with a point ${\alpha\in{\mathbb C}}$ and apply a positive word in ${f}$ and ${g}$. Get

$\displaystyle \begin{array}{rcl} 1\pm c\pm c^2 \pm c^3+\cdots+\pm c^{n-1}+c^n \alpha. \end{array}$

This converges to ${F(c)}$, ${F}$ a ${\pm1}$ series. If ${f\Lambda\cap g\Lambda\not=\emptyset}$, a ${\{-1,0,1\}}$-series vanishes on ${c}$.

A computer picture shows that ${M}$ looks like an annulus with outer boundary the unit circle and a hairy inner boundary with two whiskers: segments of the real axis. Calegari-Koch-Walker: the interior of ${M}$ is dense in ${M}$ away from the real axis. This helps drawing certified pictures. Play with our software! ${M}$ turns out to have infinitely many holes. We conjecture that holes accumulate on every point of the frontier of ${M}$ (away from the real axis). We also conjecture that algebraic points of the frontier are dense there.

Questions.

1. How do cutpoints of ${\Lambda}$ depend on ${c}$ ?
2. Give a simple topological model for the action of ${G}$ on ${\Lambda}$.

1.2. Uniformization

When ${c\in M}$, uniformize the complement of ${\Lambda}$, yeilds a parametrization of ${\partial \Lambda}$. For ${c\in\partial M}$, the dynamics on ${\partial\Lambda}$ can be partially lifted to the circle. It can be merged into a single piecewise continuous map ${H:S^1\rightarrow S^1}$ that preserve the dynamical lamination.

Theorem 1 (Calegari-Walker) A pair of points of ${S^1}$ corresponds to a cut-point of ${\Lambda}$ iff all iterates of ${H}$ are defined on it.

This leads to a criterion for the existence of cut-points, in terms of the dynamical lamination. This is algorithmic: a directed graph is inductively produced, cut-points correspond to infinite directed paths in it. Walk on this graph amounts to dynamics on the set of cut-points.

The action of of ${H}$ on ${S^1}$ is conjugate to a (discontinuous) piecewise linear action.

1.3. Dendrites

When the stretch of ${H}$ is equal to 2, ${\Lambda}$ is a dendrite, ${H}$ is quasiconformally conjugate to degree 2 rational maps studied by Bandt, Solomyak.

2. Questions

What Hausdorff dimension? Easy: cut into two pieces, apply dilation by ${c}$.

What if ${c}$ is algebraic? Analogue to Misiurewicz points.