The billiard table with a nowhere differentiable boundary is not well defined; the law of reflection holds a no point of the boundary.  Denoting the Koch snowflake by KS, the billiard Omega(KS) is a canonical example of such a table and the focus of the talk.  We will show that KS being approximated by a sequence of rational polygons and Omega(KS) being tiled by equilateral triangles both allow us to construct what we call a sequence of compatible periodic hybrid orbits.  Under certain situations, such sequences have interesting limiting behavior indicative of the existence of a well-defined billiard orbit of Omega(KS).  In addition to this, we provide a topological dichotomy for a sequence of compatible orbits.  Other important properties and interesting results will be discussed, especially with regards to the possible presence of self-similarity in what we propose to be a well-defined periodic hybrid of the Koch snowflake fractal billiard Omega(KS).  Finally, we will briefly discuss future research problems.

Motivated by the theory of fractal strings and complex dimensions of M. L. Lapidus and M. van Frankenhuijsen, we define a class of fractal strings for self-similar measures based on scaling regularity. In turn, these fractal strings allow for an analysis of the symbolic dynamics on such measures via the abscissae of convergence of scaling zeta functions. With this approach, we recover (among other things) Moran's theorem regarding the Hausdorff dimension of self-similar sets and the Hausdorff dimensions of Besicovitch subsets.

We will discuss several open problems on dynamical properties of interval translation mappings. In particular, we will observe and discuss an interesting change in the limiting behavior in the case when some randomness is added to the system. 

In the previous two talks we established a dictionary between some properties of quasiperiodic (particularly Fibonacci) models and some geometric constructions arising as dynamical invariants for the Fibonacci trace map. In this talk we shall apply our findings to a specific model: the classical 1D Ising model with quasiperiodic magnetic field and quasiperiodic nearest neighbor interaction. In particular, we'll prove absence of phase transitions of any order and we'll investigate the structure of Lee-Yang zeroes in the thermodynamic limit (these are zeroes of the partition function as a function of the complexified magnetic field---while in finite volume the partition function is a polynomial whose zeroes fall on the unit circle, a challenge is to determine whether in infinite volume (thermodynamic limit) these zeroes accumulate on any set on the unit circle, and if so, to determine the structure of this set). The purpose of this work is to serve as rigorous justification to previously observed phenomena (mostly through numerical and some soft analysis). Should we have time, we'll also very briefly mention applications of the aforementioned dictionary to quasiperiodic Jacobi matrices/CMV matrices.

Over the past almost three decades dynamical systems have played a central role in spectral analysis of quasiperiodic Hamiltonians as well as certain quasiperiodic models in statistical mechanics (most notably: the Ising model, both quantum and classical). There are many ways of introducing quasiperiodicity into a model. We shall concentrate on the widely studied Fibonacci case (which is a prototypical example of so-called substitution systems on two letters with certain desirable properties). In this case a particular geometric scheme, arising from a certain smooth three-dimensional dynamical system associated to the quasiperiodic model in question (the so-called Fibonacci trace map) has been established. Our aim is to present a general dynamical/geometric framework and to demonstrate how information about the model in question (spectral properties for Hamiltonians, and Lee-Yang zeros distribution for classical Ising models) can be obtained from the aforementioned dynamical system and the geometry of certain dynamically invariant sets. In this first in a series of two (or three) talks, we'll briefly recall how dynamical systems are associated to Schroedinger and Jacobi operators, as well as classical Ising models. We'll establish notation, ask main questions and in general prepare the ground for a somewhat more general (in terms of geometry and dynamical systems) discussion for next time.