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 with the same stretch , , but distinct fixed points. E.g. and . Let be the semi-group they generate. It has a limit set , a compact set which is forward invariant under .
Easy to draw: start with a large disk, take its image by the -ball of (a union of little disks), for large (indeed, is the intersection of these as tends to infinity).
The Barnsley-Harrington Mandelbrot set is the set of parameters for which the is connected. Equivalently if . Equivalently, for which is a root of a power series with coefficients in . Equivalently, for which is a limit of roots of polynomials with coefficients in .
How do roots arise ? Start with a point and apply a positive word in and . Get
This converges to , a series. If , a -series vanishes on .
A computer picture shows that 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 is dense in away from the real axis. This helps drawing certified pictures. Play with our software! turns out to have infinitely many holes. We conjecture that holes accumulate on every point of the frontier of (away from the real axis). We also conjecture that algebraic points of the frontier are dense there.
- How do cutpoints of depend on ?
- Give a simple topological model for the action of on .
When , uniformize the complement of , yeilds a parametrization of . For , the dynamics on can be partially lifted to the circle. It can be merged into a single piecewise continuous map that preserve the dynamical lamination.
Theorem 1 (Calegari-Walker) A pair of points of corresponds to a cut-point of iff all iterates of 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 on is conjugate to a (discontinuous) piecewise linear action.
When the stretch of is equal to 2, is a dendrite, is quasiconformally conjugate to degree 2 rational maps studied by Bandt, Solomyak.
What Hausdorff dimension? Easy: cut into two pieces, apply dilation by .
What if is algebraic? Analogue to Misiurewicz points.