Minicourse

• Aaron Brown (Northwestern University, USA)

Stationary measures fo rdiscrete random dynamical systems

I will discuss rigidity properties of stationary measures for actions of large groups. I’ll focus on questions of stiffness, classification, absolute continuity, the SRB property, and their relationship to
classification of orbit closures. I will primarily discuss results on surfaces with some discussion on workin progress in higher-dimensions.

• Balázs Bárány (Budapest University of Technology and Economics, Hungria)

The transversality method and its applications

The “transversality method” has been developed to study parametrized families of fractal sets and measures. In this minilecture, we will give a brief overview of the classical and most recent results. We will then study the method’s applications on orthogonal projections, iterated function systems, and stationary measures to study dimension, absolute continuity, and further regularity properties and we will verify the
condition for several certain cases.

• Viveka Erlandson (University of Bristol, UK)

Hyperbolic surfaces and Mirzakhani’s curve counting

It’s aclassical problem to study the growth of the number of periodic orbits of bounded length L under a geodesic flow on a manifold. When it comes to (closed) hyperbolic surfaces Huber proved that the number is a symptotic to eL /L and this results has since been generalized in many directions, maybe most famously by Margulis to negatively curved manifolds as well as more general flows. However, a fundamentally different result is due to Mirzakhani who counted the subset of those closed geodesics in hyperbolic surfaces which have no self-intersections (or more generally, those inside a fixed mapping class group orbit) and showed that in this setting the growth is asymptotic to a polynomial in L of degree only depending on the topology of the surface. In this mini-course we will give the necessary background on hyperbolic surfaces to understand these results, give an idea of the proof of Mirzakhani’s theorem and introduce some tools used such as measured laminations and train tracks. Time allowing we present some generalizations (for example, to non-simple closed geodesics or to various metrics) and applications to the distribution of closed geodesics.