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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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