Any (at most countable) family of pairwise transverse (even singular, with saddle or prong singularities) foliations on the plane R² admits a compactification (as the disc D²) by a circle at infinity so that every ray in a leaf tends to a point on the circle, and this compactification is unique up to two natural requirements. Thus every leaf corresponds to a pair of points on the circle. With Th. Barthelmé and K. Mann, we consider the reverse problem and we give a complete answer to the two following questions:

Q1: (realization) Under which hypotheses two sets L⁺, L⁻ of pairs of points on the circle are precisely the pairs of endpoints of leaves of two transverse foliations (we give the answer for singular, and also for nonsingular foliations, and we prove that the foliations are uniquely determined).

More important is the second question:
Q2: (completion) Under which hypotheses two sets l⁺,l⁻ or pair of points on the circle correspond the pairs of endpoints of a dense subset of leaves of two transverse foliations (singular or nonsingular, and uniqueness). The uniqueness implies that any group action on the circle preserving l⁺,l⁻ extends in an action on the disc preserving the corresponding foliations. This allows us to prove that if an action G -> Homeo_+(S¹) of a group on the circle is induced by an Anosov-like action G -> Homeo_+(D²), then this action is unique and completely determined by the action on the circle. With Th. Marty, we consider the case of 1 (singular or not) foliation and we give a complete answer to the following questions

Q3: (realization) Under which hypotheses a set L of pairs of points on the circle is precisely the set of pairs of endpoints of leaves of a foliation,

Q4: (completion) Under which hypotheses a set l of pairs of points on the circle corresponds to the pairs of endpoints of a dense subset of leaves of a foliation and we prove again the uniqueness.[-]

Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond.[-]

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.[-]

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.[-]

Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond.[-]

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.[-]

A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\'e return maps of the periodic orbits, volume Lyapunov exponents. After a brief overview of some classical results we will focus on recent developments in rigidity of Anosov and partially hyperbolic systems as well as connections to geometric rigidity. The latter is based on joint work with B. Kalinin and V. Sadovskaya and with F. Rodriguez Hertz.[-]