# Abstracts & Slides

### Damien Gaboriau

Title: On dense totipotent free subgroups in full groups
Abstract: I will present the following notions and results.The perfect kernel of a countable group Gamma is the largest closed subspace of the space of subgroups of Gamma without isolated points. We introduce the class of totipotent ergodic probability measure preserving (p.m.p.) actions of Gamma: those for which almost every point-stabilizer has dense orbit in the perfect kernel (under conjugation). Equivalently, the support of the associated Invariant Random Subgroup (IRS) is as large as possible, namely it is equal to the whole perfect kernel. We prove that for every non-free action of the free group $$F_r$$ on $$r$$ generators, there exists another action of $$F_r$$ that has the same orbits and that is totipotent. Moreover, the image in the associated measured full group $$[R]$$ is dense (this provides plenty of additional nice properties). We observe that these actions have no minimal models.This also provides a continuum of pairwise orbit inequivalent IRS of $$F_r$$, all of whose supports are equal to the whole space of infinite index subgroups.
This is joint work with Alessandro Carderi and François Le Maître.

### Nathalie Aubrun

Title: Subshifts of finite type on amenable Baumslag-Solitar groups
Abstract: Amenable Baumslag-Solitar groups are two generators one relator groups with presentation $$\langle a,t | at=ta^n \rangle$$ with $$n$$ a positive integer. In the first part of this talk I will present in details these groups and their Cayley graph, and how to embed them in $$\mathbb{R}^2$$. I will then review recent results about subshifts of finite type (SFTs) on amenable Baumslag-Solitar groups: indecidability of the emptiness problem for SFTs and existence of aperiodic SFTs. Based on joint works with Jarkko Kari and with Michael Schraudner.

### Corinna Ulcigrai

Title: Geometric rigidity in genus two
Abstract: We say that a class of dynamical systems is geometrically rigid if a topological conjugacy between two elements in this class is necessarily differentiable. A very important class of geometrically rigid
dynamical systems are circle diffeomorphisms with Diophantine rotation number, or, correspondingly, foliations on tori (i.e. genus one surfaces). In this short talk we will advertise a generalization of this result to foliations on surfaces of genus two and their Poincaré maps, namely generalized interval exchange transformations. This is joint work with Selim Ghazouani.

### Aimee Johnson

Title: Speedups of higher dimensional actions.
Abstract: Speedups in the measurable setting have been around for decades, but only recently has the topological setting been explored. In this talk I will discuss joint work with David McClendon in topological speedups of higher dimensional actions. In other words, we will consider systems $$(X,T)$$ where $$X$$ is a Cantor space and T is a minimal $$\mathbb{Z}^d$$-action. Our goal is to investigate how speedups of these higher dimensional actions are similar to, or different from, the single transformation scenario. Particular attention will be paid to the example of an odometer, so we’ll review what a $$\mathbb{Z}^d$$ odometer is and think about what a speedup of it might look like.

### John Johnson

Title: Relative notions of syndetic and thick sets and their composition
Abstract: Syndetic and thick subsets of positive integers are two classical “notions of size” that appear in both dynamics and combinatorics related to Ramsey theory. Building on a generalization of syndetic sets due to Shuungula, Zelenyuk, and Zelenyuk, we give a generalization of thick sets and demonstrate how these notions can be “composed” to produce both new and old notions of size. Moreover, these compositions appear to mimic certain aspects of classical combinatorial constructions in Ramsey theory and suggest a classification problem that roughly asks how various compositions are related. I’ll illustrate these notions via a suggestive visualization and indicate some possible connections with classifying dynamical systems. (Based on a joint project with Cory Christopherson and Florian Richter.)

### Cristobal Rojas

Title: The computational complexity of describing the asymptotic behavior of real quadratic maps
Abstract: One of the main goals of dynamical systems theory is t
he classification and description of the asymptotic behavior of typical trajectories. The real quadratic family is an example where this classification is essentially complete in terms of topological descriptions (attractors) as well as statistical (physical measures). In this talk we will review recent results complementing this classification in terms of the computational complexity of describing these asymptotic objects. This is joint work with M. Yampolsky.