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# Workshop "Dynamics, Geometry and Groups": Abstracts

## Mario Bonk

### Quasisymmetric rigidity for Sierpinski carpets

Sierpinski carpets exhibit surprising rigidity under quasisymmetric maps. This phenomon appears in various contexts in geometric group theory and complex dynamics. In my talk I will give a survey of some recent results in this area.

## Dzmitry Dudko

### Conjugacy problem in a mapping class biset

Given a Thurston map $f$, its mapping class biset $M(f)$ is the set of maps obtained by pre- and post-composing $f$ with elements in the mapping class group. Maps in $M(f)$ are considered up to isotopy relative the postcritical set. Two maps in $M(f)$ are conjugate precisely when they are Thurston equivalent. We will discuss how to do computations in $M(f)$. In particular, we will show that the conjugacy problem in $M(f)$ is decidable.

Based on a joint work Laurent Bartholdi.

## Mikhail Hlushchanka

### Invariant graphs, tilings, and iterated monodromy groups

One of the main open problems in holomorphic dynamics is to obtain “nice” combinatorial models for rational maps and classify the maps in combinatorial terms. Such models were constructed for postcritically-finite polynomials by Douady and Hubbard in the 1980’s. However, the case of general rational maps is much more complicated and still draws lots of attention. I will describe combinatorial models given by invariant planar embedded graphs for different classes of rational maps. I will also discuss how these graphs, and the respective tilings of the Riemann sphere, can be used to study the properties of iterated monodromy groups.

## Daniel Meyer

### When is a map rational without using Thurston’s theorem.

Thurton’s celebrated characterization of rational maps gives a criterion when a Thurston map (i.e., a topological analog of a rational map) is equivalent to a rational map. If a Thurston map $f$ is expanding in a suitable sense we may define a visual metric $\rho$ on $S^2$. The geometry of $(S^2,\rho)$ mirrors properties of the map $f$. In particular $(S^2,\rho)$ is quasisymmetric to the Riemann sphere if and only if $f$ is topologically conjugate to a rational map (a result independently obtained by Haissinsky-Pilgrim). Since the proof does not depend on Thurston’s theorem it allows to decide when $f$ is conjugate/equivalent to a rational map using different methods. While there is no general result yet, it is possible to do so in specific examples.

The talk is based on joint work with Mario Bonk.

## Volodymyr Nekrashevych

### Iterated monodromy groups and amenability

I will give a survey of known results and open questions concerning amenability of iterated monodromy groups. In particular, I will describe the largest class of rational functions for which we know that their IMGs are amenable.

## Kevin Pilgrim

### Hausdorff and conformal dimension of Julia sets

Suppose $f: \overline{\mathbb{C}} \to \overline{\mathbb{C}}$ is a rational map from the Riemann sphere to itself. Iterating $f$ gives a dynamical system. The Julia set $J(f)$ is the chaotic locus; it is typically a “fractal” set. The Hausdorff dimension $\mathrm{hdim}(J(f)) \in (0,2]$ is a crude measurement of the “thickness” of $J(f)$. The conformal dimension $\mathrm{confdim}(J(f))$ is the infimum of the Hausdorff dimension $\mathrm{hdim}(Y)$ where $Y$ is a metric space quasisymmetrically equivalent to $J(f)$. I will survey some known results about Hausdorff and conformal dimensions of Julia sets, concentrating on hyperbolic (expanding) maps.

### Numerical invariants of expanding self-covers

We generalize the setup of the previous talk to an arbitrary self-cover $f: X \to X$ of a compact, connected, locally connected topological space $X$. Well-known constructions yield a family of metrics on $X$ in which the dynamics is “conformal” in the sense that $f(B(x,r))=B(f(x),\lambda r)$ for all small balls. It follows that there exist Ahlfors regular metrics on $X$ and thus that the conformal dimension becomes a numerical invariant of the topological dynamics.

Expanding “virtual graph endomorphisms” $\pi, \phi: \Gamma_1 \to \Gamma_0$ (mentioned by D. Thurston in his lectures here) determine such dynamical systems $f: X \to X$ and so give a wealth of examples. The corresponding asymptotic energies $\overline{E}_p^p[\pi, \phi]$ then give additional numerical invariants of the dynamics. This gives new invariants of hyperbolic rational maps with connected Julia sets.

### Conformal dimension is a critical exponent of asymptotic energy

I will discuss the ingredients in the proof of the following result, which is ongoing joint work with D. Thurston. Suppose $\pi, \phi: \Gamma_1 \to \Gamma_0$ is an expanding virtual graph endomorphism with $\Gamma_1, \Gamma_0$ connected and $\phi$ surjective on fundamental groups. Then the conformal dimension of the limit dynamical system $f: X \to X$ is equal to the unique exponent $p$ for which $\overline{E}_p^p[\pi, \phi]=1$. We apply this to give explicit estimates for the conformal dimension of Sierpinski carpet Julia sets in the family $z^2+\lambda/z^2$, $\lambda<0$.

## Bernhard Reinke

### Orbital Schreier graphs for IMGs of entire functions

Transcendental entire functions have many dynamical properties analogous to polynomial maps. For post-critically finite polynomials, the iterated mondromy groups are amenable. In this talk, I address the question on how to extend this result to the realm of post-singulary finite transcendental functions, focussing mainly on a particular simple example, namely the iterated monodromy group of the function $f(z)=2\pi i e^z$.

## Dierk Schleicher

### On the dynamics of entire functions and their combinatorial characterization

The dynamics of transcendental entire functions can be very different from polynomial dynamics (the essential singularity at infinity behaves very differently from a superattracting fixed point), and yet in certain cases it can be quite similar, especially in the postsingularly finite case. We present some of the difficulties and surprises, and we present some positive results linking transcendental dynamics and symbolic dynamics via Thurston theory.

## Dylan Thurston

### Combinatorial models of rational maps

The underlying topological structure behind rational maps can be represented in multiple ways: branched self-covers of the sphere; virtual endomorphism of groups; bisets over groups; automata; groups acting on trees; virtual endomorphisms of graphs. We explain how these different representations are related to each other, with particular emphasis on graphs.

### Elastic graphs and other graph energies

We describe the energies $E^p_q$, for $1 \le p \le q \le \infty$, for maps between graphs. Special cases, for $p,q \in \{1,2,\infty\}$, include length of a curve, Lipschitz stretch factor, Dirichlet energy, and extremal length. The energy $E^2_2$, which has not been previously considered, captures when one elastic network is “looser” than another for all possible targets.

### Positive characterization of rational maps

We give a positive characterization of hyperbolic rational maps, a certificate that guarantees that a branched self-cover of the sphere is equivalent to a rational map, based on comparisons between elastic networks. This is complementary to W. Thurston’s earlier theory of obstructions.