Second meeting, Tel Aviv University, 29.12.22
Sullivan's Dictionary, Limits of deformations, and Modular Laminations
Sullivan's dictionary between Kleinian groups and rational maps reveals how many objects, such as limit sets and Julia sets, are different names for the same thing. On a deeper level, it provides conjectures in one field that are analogs of well-known theorems in the other. One such well-known theorem, proven by W. Thurston, is the compactness of the space of representations (in Isom(H^3)) of the fundamental group of a compact 3-manifold with acylindrical boundary. Because there is no quotient three-manifold for a rational map, new tools are needed to study degenerating sequences of deformations, so we introduce the concept of an invariant modular lamination and use it to prove the analog of this theorem of Thurston.
Hyperbolic tilings and their orbits
In my talk I will introduce a recent construction of tilings of the hyperbolic upper half-space H^{d+1}. These tilings may be viewed as extensions of hyperbolic tilings considered by Boroczky, Penrose and Kamae, and essentially illustrated by Escher, as well as hyperbolic liftings of the d-dimensional Euclidean multiscale substitution tilings introduced in our earlier work. I will describe our results about the geodesic and horospheric actions on the associated space of hyperbolic tilings and discuss a prime orbit theorem for the geodesic flow. Based on joint work in progress with Yaar Solomon.
Hyperuniformity and non-hyperuniformity of mathematical quasicrystals
Meyer's cut-and-project method allows one to construct mathematical quasicrystals, i.e. pure-point diffractive point processes, starting from an arbitrary lattice and "window". This talk is concerned with the question how finer diophantine properties of the underlying data are reflected in the diffraction measures of these point processes. It has long been believed (and been observed experimentally) that, compared to Poisson processes, mathematical quasicrystals have reduced long wavelength density fluctuations, a phenomenon known as "hyperuniformity", which spectrally corresponds to a certain decay of the diffraction measure around the trivial character. We provide the first rigorous proof that this is indeed the case for generic quasicrystals in Euclidean space (with Fourier smooth window) and at the same time provide the first examples of "anti-hyperuniform" quasicrystals. These counterexamples are based on quasicrystals whose underlying parameters are Liouville numbers. Joint work with Michael Bjoerklund.
A CLT for non-uniformly expanding random dynamical systems under explicit conditions
The central limit theorem (CLT) for (partially) expanding or hyperbolic dynamical systems was extensively studied in the past decades (including several quantitative versions etc.) A random dynamical system (RDS) is formed by compositions of random stationary maps along orbits of a "driving system" (MPS). Limit theorems for RDS have been studied extensively in the past years for uniformly expanding RDS, or when the maps involved are independent. We present the first types of explicit conditions for the CLT (and related results) for random non-uniformly expanding dynamical systems, which are not driven by an iid sequence. Our approach is based on proving an "effective" random Perron-Frobinus "rate" for the products of the underlying random transfer operators together with certain weak upper mixing type assumptions on the driving system.
Nonpositive immersions
A 2-complex X has nonpositive immersions if for every immersion Y --> X with Y compact and connected, either euler(Y)\leq 0 or \pi_1(Y)=1. I will give a survey on nonpositive immersions and some variants definitions, and discuss the relationships with various properties of X and subgroup properties of \pi_1(X). I'll describe some of the conjectures in the topic and some recent progress.
You are cordially invited.