# Mini-courses

• K. BURNS & B. SCHAPIRA
• Title: “Geodesic flows in nonpositive curvature”
• Abstract: We will discuss ergodic properties of geodesic flows in nonpositive curvature, with emphasize on the differences with the negative curvature case, where geodesic flows are uniformly hyperbolic. We will first give some geometric background on nonpositively curved manifolds, allowing to understand where key dynamical properties (closing lemma, product structure) come from. We will present results due to Coudène-Schapira on generic properties of invariant measures, and to Burns-Gelfert/Gelfert-Schapira on thermodynamical formalism.
We will conclude with some open questions in the subject.

• L. DIAZ & A. GORODETSKI
• Title: “Nonhyperblic ergodic measures”
• Abstract: Instead of the set of diffeomorphisms that preserve a Lebesgue (or some other fixed) measure, which is the standard setting in smooth ergodic theory, one can consider the set of all invariant ergodic measures for a given map. It is reasonable to expect that many of the properties of a map should be reflected in terms of the properties of its invariant measures. For example, a local diffeomorphism with positive Lyapunov exponents for every invariant ergodic measure must be uniformly expanding [AAS, C]. As another example, if for some diffeomorphism all the atomic measures of all its C 1 -perturbations have only non-zero Lyapunov exponents, then the diffeomorphism must satisfy Axiom A [A, H]. In [DG] we conjectured that a generic diffeomorphism must either be uniformly hyperbolic or exhibit a non-hyperbolic (having some zero Lyapunov exponents) invariant ergodic measure. This question is closely related to the question on connectedness of the space of ergodic invariant measures of a given diffeomorphism. Currently there are two methods of construction of ergodic non-hyperbolic measures in smooth dynamics. First, the method based on periodic approximations was introduced in [GIKN] in the setting of skew-products, and later used to construct open sets of diffeomorphisms with such measures [KN] and applied to generic non-hyperbolic homoclinic classes of diffeomorphisms [DG, BDG]. This approach provides conditions for a sequence of atomic measures to converge to a non-trivial non-hyperbolic ergodic measure. Second, in [BBD] some C 1 -open conditions were stated that guarantee that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. These conditions are satisfied for a large class of non-hyperbolic $C^1$ diffeomorphisms, and imply existence of a partially hyperbolic compact set with onedimensional center direction and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The method uses a construction of a blender defined dynamically in terms of strict invariance of a family of discs, and allows to construct a $C^1$ -open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with non-hyperbolic ergodic measures with positive entropy. In the mini-course we intend to discuss both types of technics, and mention many related open questions.

• A. ESKIN & A. WRIGHT
• Title: “The $SL(2,\mathbb{R})$ action on moduli space”
• Abstract: The first part of the mini-course will introduce translation surfaces and the $GL(2,\mathbb{R})$ action on moduli spaces of translation surfaces. The second part of the course will deal with random walks on homogeneous spaces $G/ \Gamma$, and will present a variant of a proof of a theorem of Benoist and Quint on stationary measures. This introduces ideas used to prove measure rigidity in the translation surface case.

• V. KLEPTSYN
• Title: “Actions on the circle: what do we know, and what should we know?”
• Abstract: My talk will be devoted to the understanding of finitely generated groups acting (sufficiently smoothly) on the circle.
There are long-standing questions in this domain, asked by Ghys, Sullivan and Hector in the 80s:if the action of a finitely generated group $G$ is minimal, is it Lebesgue-ergodic? If there is a Cantor minimal set $\Lambda$, is this set necessarily of Lebesgue measure zero? Are then there only finitely many orbits of $G$-action on the set of connected components of $S^1\setminus \Lambda$?
These questions can be also naturally stated for non-singular codimension one foliations of compact manifolds.

I will speak on recent progress, achieved in joint works with Bertrand Deroin, Dmitry Filimonov, Andrés Navas, Michele Triestino, Sebastian Alvarez, Dominique Malicet, Carlos Meniño. Not only we have succeeded in establishing positive answers on the questions above for the case of the analytic actions, except for two sub-cases that are still to be treated. More importantly, a new understanding seems to emerge: a better dichotomy is not « minimal actions vs exceptional minimal set », but « locally non-discrete actions vs actions admitting a Markov partition ». The latter is still to be established in full generality, but it seems a perfect goal for an attack: once it is established, full characterization of such actions and their properties become a natural corollary.

• A. KOROPECKI & A. KOCSARD
• Title: “Rotation theory of torus homeomorphisms”
• Abstract: The main focus of this mini-course is to introduce some tools and techniques which have been extensively used in recent results in topological surface dynamics. We will then use them to study rotational deviations of two-dimensional toral homeomorphisms in the isotopy class of the identity, in particular in the case where the rotation set has empty interior.

• K. MANN
• Title: “Rigidity and flexibility for groups acting on the circle”
• Abstract:  An action of a group G on the circle is rigid if every deformation of the action is as trivial as possible — in this case, the right notion is “semi-conjugate”.  In this series of talks, I’ll introduce and contrast techniques available to study rigidity of group actions on the circle in various settings ($C^0, C^1$, smooth…) with a focus on new perspectives and tools for understanding groups acting on the circle by homeomorphisms.

• I. MELBOURNE
• Title: “Averaging and homogenization for deterministic fast-slow systems”
• Abstract: I will describe how stochastic differential equations arise as limits of deterministic dynamical systems.   In particular, a classical question in stochastic analysis — the correct interpretation of stochastic integrals — is given a definitive answer.
The techniques range from smooth ergodic theory and basic probability theory to cutting-edge stochastic analysis in the form of rough path theory.