Analysis Aspects of Dynamics

Scaling limits of random structures in two dimensions often posses some conformal invariance properties, giving ways to methods of geometric analysis.

Among the fascinating questions here are the configurations of random tilings under scaling limits, and the boundaries between their ordered and disordered, or liquid, limit regions. The liquid domains carry a natural complex structure, which turns out can be described by a quasilinear Beltrami equation with very specific properties.

In this talk, based on joint work with E. Duse, I. Prause and X. Zhong, I show how this approach leads to understanding and classifying the geometry of the frozen boundaries for different random tilings and other dimer models.

Geometric structure of Laplace eigenfunctions allows to distinguish between systems with integrable and chaotic 'quantum billiards'. It was proposed by M.Berry that in the chaotic case the corresponding high-energy wave functions (eigenfunctions of the Laplacian) could be modelled by a random plane wave model. Random plane wave could be informally describes as a random superposition of all plane waves with the same energy. In 2001 Bogomolny and Schmit proposed that the geometric structure of the level sets of the random plane wave could be described by the square lattice bond percolation. In the recent years it became apparent that this is a part of a wider picture: many random fields exhibit the same behaviour and a their large scale behaviour is described by percolation models. In this the talk I will present several recent results that explore the connection between random fields and percolation.

Linear dynamics has been a rapidly evolving area since the early 1990s. It lies at the intersection of operator theory and topological dynamics, and its central property is hypercyclicity. In the first part of this talk I will give a general introduction to linear dynamics and I will demonstrate that many natural continuous linear maps turn out to be hypercyclic.

The second part of the talk is concerned with the stronger property of frequent hypercyclicity, which was introduced by Bayart and Grivaux in 2004, and has fundamental connections to ergodic theory. In particular I will outline some recent results on the sharp growth rates, in terms of the $L^2$-norm on spheres, of harmonic functions that are frequently hypercyclic with respect to the partial differentiation operator. This answers a question posed by Blasco, Bonilla and Grosse-Erdmann (2010).

This is joint work with Eero Saksman and Hans-Olav Tylli.

We give an ad-hoc definition of a non-uniformly hyperbolic flow on a smooth manifold (or orbifold). The Teichmüller flow on a stratum of abelian or quadratic differentials as well as the geodesic flow on a rank one manifold fulfil this property. We then show how this property can be used to obtain an equidiistribution result for periodic orbits as well as a simplicity result for the Lyapunov spectrum of a flat cocycle over the flow with respect to the unique measure of maximal entropy.

Motivated by the theories of Kleinian groups and of continued fractions we explore the dynamics of semigroups generated by hyperbolic isometries of the unit 3-ball. Each Kleinian group induces a limit set contained in the Riemann sphere. In an analogous fashion each semigroup induces two subsets of the Riemann sphere, a forward limit set and a backward limit set. We describe semigroups for which the identity is not an accumulation point as semidiscrete. We have shown that if S is a finitely-generated nonelementary semidiscrete semigroup of isometries of the unit 2-ball, then S is a group if and only if its two limit sets are equal. In this talk we show an extended version of that result to semigroups acting on the unit 3-ball.

Uniformly quasiregular (UQR) maps are a generalization of holomorphic dynamics to higher dimensional quasiconformal geometry. While it is well known which closed 2D surfaces admit non-injective holomorphic dynamics, the same question of characterizing existence for non-injective UQR maps $f$ on closed Riemannian manifolds $M$ remains open. An approach using Sobolev cohomology has yielded new obstructions, including restrictions on the degree of $f$, a sharp upper bound on the dimension of the cohomology ring $H^*(M; \mathbb{R})$, and positive measure of the Julia set of $f$ if $M$ is not a rational homology sphere. Parts of the talk are based on joint work with Pekka Pankka.

The almost Mathieu operator is the central object of study in the theory of quasiperiodic Schroedinger operators. We review results on the properties of its spectrum (which is a Cantor set). For frequencies admitting a power-law approximation by rationals, we show that the central gaps of H are open and provide a lower bound for their widths.

Benjamini, Schramm and Timar introduced the notion of "separation profile" for a graph or finitely generated group. Hume, Tessera and I generalised this notion to define an invariant called the Poincaré profile of a graph or group; roughly speaking, this measures which discrete Poincaré inequalities hold on subsets of the graph. In this talk I will focus on Gromov hyperbolic groups and explain the connection between the Poincaré profile and Pansu's conformal dimension of the boundary at infinity, and applications of this to non-embedding results. Joint work with Hume and Tessera.

A quasisymmetric map is one that changes angles in a controlled way. As such they are generalizations of conformal maps and appear naturally in many areas, including Complex Analysis and Geometric group theory. A quasisphere is a metric sphere that is quasisymmetrically equivalent to the standard $2$-sphere. An important open question is to give a characterization of quasispheres. This is closely related to Cannon's conjecture. This conjecture may be formulated as stipulating that a group that ``behaves topologically'' as a Kleinian group ``is geometrically'' such a group. Equivalently, it stipulates that the ``boundary at infinity'' of such groups is a quasisphere.

A Thurston map is a map that behaves ``topologically'' as a rational map, i.e., a branched covering of the $2$-sphere that is postcritically finite. A question that is analog to Cannon's conjecture is whether a Thurston map ``is'' a rational map. This is answered by Thurston's classification of rational maps.

For Thurston maps that are expanding in a suitable sense, we may define ``visual metrics''. The map then is (topologically conjugate) to a rational map if and only if the sphere equipped with such a metric is a quasisphere. This talk is based on joint work with Mario Bonk.

Dynamics of a uniformly quasiregular mapping can be viewed as a conformal dynamics with respect to a (non-standard) measurable conformal structure. Two typical constructions of uniformly quasiregular mappings are the conformal trap construction of Iwaniec and Martin and the Lattès construction. In this talk I will discuss a construction associated to a quasi-self-similar wild Cantor set on the 4-sphere. This is joint work with Jang-Mei Wu.

We will consider a generalization of the Selberg zeta function for cocompact Fuchsian groups in the broader context of higher Teichmüller theory. In particular, we will consider the extension of this function to the entire complex plane and the location of poles/zeros.

I will review instances of holomorphic interpolation in the context of quasiconformal mappings. This technique can be used to upgrade natural a priori bounds by exploiting holomorphic dependence through parameter space. The resulting estimates are usually powered by (a variant of) the Schwarz lemma. Topics include variation of dimension of Julia sets, quasiconvexity of energy functionals and twisting estimates for bilipschitz and conformal maps.

The classical conformal welding problem consists of constructing a Jordan curve as the result of glueing two conformal discs along their boundaries. We will discuss the analogous problem that leads to dendrites, and present applications in complex dynamics and in probability theory.

Given a measure preserving action of a group $G$ on a probability space $X$ and a real valued function $f$ on $X$, we consider the spherical averages $S_n(f)$ of the functions $f(g.x)$ averaged over all elements $g$ of length $n$ in a fixed set of generators.

The limiting behaviour of $S_n(f)$ has long been studied. Cesaro convergence has been proved in a wide variety of contexts. Actual convergence (depending on the parity of $n$) for free groups was proved by Nevo-Stein for $f$ in $L^p$, $p>1$

In 2002, Bufetov extended the Nevo-Stein result to $f$ in a slightly wider class by using a certain self-adjointness property of an associated Markov operator, which in turn depends on the fact that the inverse of a reduced word in a free group is itself reduced.

In this talk we announce the same result for a large class of Fuchsian groups with presentations whose relations all have even length. The method relies on a new twist on the Bowen-Series coding for Fuchsian groups: by encoding the set of all shortest words representing a particular group element simultaneously, we obtain a suitable self-adjointness property of the associated Markov operator to which we apply a variant of Bufetov's original proof.

This is joint work with Alexander Bufetov and Alexey Klimenko.

This talk concerns two longstanding conjectures concerning the iteration of transcendental entire functions. Eremenko’s conjecture is that all the components of the escaping set (the set of points which escape to infinity under iteration) are unbounded. Baker’s conjecture is that, for functions of small growth, there are no unbounded components of the Fatou (or stable) set. We discuss the history of these two apparently unrelated conjectures and how they can be tackled by using a common approach. This approach uses and extends several techniques from complex analysis including extremal length arguments to prove winding properties of level curves, growth arguments using results of Beurling, and results on the distribution of zeros building on results of Cartwright.

This is joint work with Phil Rippon and Dan Nicks.

This talk is based on joint work with Genadi Levin and Weixiao Shen, in which we develop a general approach which shows that critical relations of families of locally defined holomorphic maps on the complex plane unfold transversally in a “positively oriented” way. An important feature of this method is that the maps we consider are only required to be defined and holomorphic on a neighbourhood of some finite set.

We will illustrate this approach to obtain transversality for a wide class of one-parameter families of interval and circle maps, for example maps with flat critical points, Arnol’d circle maps, but also for many families of maps with complex analytic extensions such as certain families of polynomial-like maps.