In this course we will focus on dynamical systems whose phase space is a Cantor set. The most famous examples (but not the only ones) of such systems are the subshifts on finite alphabets. We will introduce the concept of ordered Bratteli diagram, which is a powerful tool to represent a big class of systems on the Cantor set. Using a Bratteli diagram, we can study important algebraic invariants associated to minimal Cantor systems as the dimension group and the topological full group. If we have time, we will explore the context of more general group actions on the Cantor set (beyond Z-actions).

We will talk about recent progress in the study of (topological) full groups of etale groupoids. We will discuss their normal structure, generating sets, their classification up to isomorphism, etc.. We will also talk about new examples of torsion groups, amenable groups, and groups of intermediate growth constructed using etale groupoids.

In this talk, I will speak about the mapping class group of the plane minus a Cantor set. After recalling several dynamical contexts in which this group naturally appears, including group actions on surfaces and complex dynamics, I will present the 'ray graph', which is a Gromov-hyperbolic graph on which this big mapping class group acts by isometries. If time allows, I will give a description of the Gromov-boundary of the ray graph in terms of long rays in the plane minus a Cantor set. This involves joint work with Alden Walker and work in progress with Danny Calegari, Mary He, Sarah Koch and Alden Walker.

The speed of a random walk is the average distance from its starting point in function of time. Given an arbitrary (regular) function between diffusive and linear, we construct a group (and a probability) with this speed function up to multiplicative constant. This is a joint work with Tianyi Zheng.

I will discuss new rigidity and rationality phenomena (related to the phenomenon of Arnold tongues) in the theory of nonabelian group actions on the circle. I will introduce tools that can translate questions about the existence of actions with prescribed dynamics, into finite combinatorial questions that can be answered effectively. There are connections with the theory of Diophantine approximation, and with the bounded cohomology of free groups. A special case of this theory gives a very short new proof of Naimi's theorem (i.e. the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces. This is joint work with Alden Walker.

One of the most simple families of dynamical systems are rotations on compact abelian groups. Dynamically these can be characterized using the concept of equicontinuity. We will show a classification of group extensions by characterizing natural extensions using weak forms of equicontinuity.

Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups), and discuss in particular the case of surface groups.

In this talk I will present L1 full groups, which are the measurable analogue of topological full groups. They are Polish groups which provide a complete invariant for measure-preserving invertible ergodic transformations up to flip-conjugacy. After establishing some of their basic properties, I will explain why the L1 full group of a measure-preserving ergodic invertible transformation is topologically finitely generated if and only if the transformation has finite entropy.

A subgroup H of a group G is distorted if the inclusion of H into G is not a quasi-isometric embedding. In transformation groups (e.g. groups of homeomorphisms and diffeomorphisms) subgroup distortion can place strong constraints on the dynamics of the action. In this talk I will discuss some interesting examples of transformation groups in which every cyclic subgroup is distorted; and the related concepts of strong boundedness, distortion, and uncountable cofinality. This is joint work with F. Le Roux.

Let G be a countable group. The space of subgroups of G, endowed with the Chabauty topology, is naturally a compact space on which G acts continuously by conjugation. The talk will focus on the topological dynamics of this action, in particular on the study of its minimal invariant subsets (named uniformly recurrent subgroups). I will explain a method to study the uniformly recurrent subgroups of a class of groups of homeomorphisms, such as Thompsonâ€™s groups and their relatives, some groups acting on rooted and non-rooted trees, topological full groups. I will discuss applications to the simplicity of the reduced C^*-algebras of these groups, linked to uniformly recurrent subgroups by results of Kennedy and Kalantar-Kennedy, and to rigidity-type results for non-free actions. This is joint work with Adrien Le Boudec.

Let K be a Cantor set contained in a real line. We call group of diffeomorphisms of K the group of homeomorphisms of K which are locally restrictions of diffeomorphisms of the line. Equivalently, if you embed the line in the plane R^2, it is the group of homeomorphisms of K which are restriction of diffeomorphisms of R^2 which preserve K. In this talk, we will discuss some properties of those groups (Burnside property, Tits alternative, distortion) and discuss their consequences on Thompson's V groups. This is a joint work with Sebastian Hurtado.

We will survey on recent joint work with C. Bleak, P. Cameron, Y. Maissel and F. Olukoya. The main goal consists in determining the automorphisms groups of these groups and relate them to dynamical issues, as it has been done in more classical settings.

The theory of Borel reducibility compares the complexity of different classification problems (given by equivalence relations on Polish spaces) and rules out certain types of complete invariants for such problems. The work I am going to describe is part of a project to understand where isomorphism of minimal flows of countable groups (and, more specifically, minimal subshifts) fits in this framework. We have two results concerning Toeplitz subshifts: one on the complexity of isomorphism of Z-Toeplitz subshifts with separated holes and one where the acting group is non-amenable (and residually finite). Those results only scratch the surface of the general problem and many interesting open questions remain. This is joint work with Marcin Sabok.