We continue with thermodynamic formalism from last time, this time applying what we have learned to hyperbolic dynamical systems. In particular, we shall see how thermodynamic formalism can be applied to obtain information about fractal dimensions of hyperbolic sets (which is, in a sense, a measure of complexity of the system). In particular, we shall cover: the Bowen's equation (relating topological pressure to Hausdorff dimension), which is a very broad generalization of Moran's theorem for some iterated function systems, as well as Ruelle's theorem (asserting that, in some sense, only observables at periodic points are needed to completely determine Bowen's equation). If time permits, we shall describe an approach, using thermodynamic formalism, to one of the open problems in spectral theory of quasiperiodic Schroedinger operators, that was presented by Anton Gorodetski at the seminar on Oct. 16th. 

We will review the inventory of open problems related to hyperbolic and partially hyperbolic dynamics (including the trace map dynamics), conservative dynamics, complex dynamics, piecewise translations, and convolutions of singular measures that are in a focus of our seminar interests (or are natural candidates for this status). Many of the problems are suitable for beginning graduate students. 
 
 

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.