The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study,this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and define the measure of maximal entropy via Patterson-Sullivan construction. In the fourth lecture, we will show that the measure of maximal entropy is unique and we will use the cross ratio function to prove the geodesic flow is mixing with respect to the constructed measure. In the last lecture, we will prove the prime geodesic theorem of the surface. This is based on two joint work with Gerhard Knieper and Vaughn Climenhaga.[-]

The aim of this lecture series is to study some ergodic properties of the geodesic flow on surfaces without conjugate points. In the first two lectures we give an introduction the geodesic flow and discuss the geometric properties of the surface that are needed to study, this includes the action of the fundamental group on the universal cover, the Gromov boundary and Morse Lemma. In the third lecture we will introduce the Poincaré series and define the measure of maximal entropy via Patterson-Sullivan construction. In the fourth lecture, we will show that the measure of maximal entropy is unique and we will use the cross ratio function to prove the geodesic flow is mixing with respect to the constructed measure. In the last lecture, we will prove the prime geodesic theorem of the surface. This is based on two joint work with Gerhard Knieper and Vaughn Climenhaga.[-]

The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is uniquely ergodic in almost every direction. We call this class of surface essentially finite.
(joint work with Anja Randecker).[-]

We prove a couple of general conditional convergence results on ergodic averages for horocycle and
geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of push-forwards of horocycle measures under the geodesic flow and a new short proof of a theorem of Chaika and Eskin on Birkhoff genericity in almost all directions for the Teichmüller geodesic flow.[-]

Over the last few years I developed (partly jointly with coauthors) dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for certain hyperbolic surfaces/orbifolds L \ H with cusps (both of finite and infinite area; arithmetic and non-arithmetic).
Both types of transfer operators arise from discretizations of the geodesic flow on L \ H. The eigenfunctions with eigenvalue 1 of slow transfer operators characterize Maass cusp forms. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. The Fredholm determinant of the fast transfer operators equals the Selberg zeta function, which yields that the zeros of the Selberg zeta function (among which are the resonances) are determined by the eigenfunctions with eigenvalue 1 of the fast transfer operators. It is a natural question how the eigenspaces of these two types of transfer operators are related to each other.[-]

These lectures are a mostly self-contained sequel to Vaughn Climenhaga's talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.[-]