We show how to construct 'simple' symbolic dynamical systems in terms of renormalisation schemes associated with multidimensional continued fractions. Continued fractions are used here to generate infinite words thanks to the iteration of infinite sequences of substitutions. Simple means that these symbolic systems have a linear number of factors of a given length, or that they have pure discrete spectrum, or else, that they have a low symbolic discrepancy. We also discuss the relation between these notions.[-]

Wang's tiles were introduced in the 1960s and have been an inexhaustible source of undecidable problems ever since. They are unit square tiles with colored edges and fixed orientation, which can be placed together provided they share the same color on their common edge. Many decision problems involving Wang tiles follow the same global structure: given a finite set of Wang tiles, is there an algorithm to determine if they tile a particular shape or subset of the infinite grid? If we look for a tiling of the whole grid, this is the domino problem which is known to be undecidable for Z2 and many other groups. In this talk we focus on infinite snake tilings. Originally the infinite snake problem asks is there exists a tiling of a self-avoiding bi-infinite path on the grid Z2. In this talk I present how to expand the scope of domino snake problems to finitely generated groups to understand how the underlying structure affects computability. This is joint work with Nicolás Bitar.[-]

For every positive integer $n$, we introduce a set $\mathcal{T}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb{Z}^2 \rightarrow \mathcal{T}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n \geqslant 1$, we consider the Wang shift $\Omega_n$ defined as the set of valid configurations over the tiles $\mathcal{T}_n$. The family $\left\{\Omega_n\right\}_{n \geqslant 1}$ broadens the relation between quadratic integers and aperiodic tilings beyond the omnipresent golden ratio as the dynamics of $\Omega_n$ involves the positive root $\beta$ of the polynomial $x^2-n x-1$. This root is sometimes called the $n$-th metallic mean, and in particular, the golden mean when $n=1$ and the silver mean when $n=2$. The family gathers the hallmarks of other small aperiodic sets of Wang tiles. When $n=1$, the set of Wang tiles $\mathcal{T}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles. The tiles in $\mathcal{T}_n$ satisfy additive versions of equations verified by the Kari-Culik aperiodic sets of 14 and 13 Wang tiles. Also configurations in $\Omega_n$ are the codings of a $\mathbb{Z}^2$-action on a 2-dimensional torus by a polygonal partition like the Jeandel-Rao aperiodic set of 11 Wang tiles. The tiles can be defined as the different instances of a square shape computer chip whose inputs and outputs are 3-dimensional integer vectors. There is an almost one-to-one factor map $\Omega_n \rightarrow \mathbb{T}^2$ which commutes the shift action on $\Omega_n$ with horizontal and vertical translations by $\beta$ on $\mathbb{T}^2$. The factor map can be explicitely defined by the average of the top labels from the same row of tiles as in Kari and Culik examples. We also show that $\Omega_n$ is self-similar, aperiodic and minimal for the shift action. Also, there exists a polygonal partition of $\mathbb{T}^2$ which we show is a Markov partition for the toral $\mathbb{Z}^2$-action. The partition and the sets of Wang tiles are symmetric which makes them, like Penrose tilings, worthy of investigation. Details can be found in the preprints available at https://arxiv.org/abs/ 2312.03652 (part I) and https://arxiv.org/abs/2403. 03197 (part II). The talk will present an overview of the main results.[-]

It has been well-known since foundational work of Hochman and Meyerovitch that that the topological entropy of a multidimensional shift of finite type may have no closed form, and in fact may even be noncomputable. For this reason, it is worthwhile to find provable approximation schemes for the entropy/pressure of "well-behaved" multidimensional models. I will describe some results guaranteeing such approximability schemes, ranging from general results requiring only mixing con-ditions on the underlying SFT to specific results tailored to individual models, and will outline some of the ways in which such results can be proven.[-]

For a class of fundamental groups of closed oriented hyperbolic 3-manifolds acting on their Gromov boundary, we compute the K-theory of the associated crossed products in terms of the first homology group of the manifold. Using classification results of purely infinite C*-algebras, we conclude that there exist infinitely many pairwise nonisomorphic torsion-free hyperbolic groups acting on their boundary, for which all crossed products are isomorphic. As in all these cases the boundary is homeomorphic to the 2-sphere, we find infinitely many pairwise non-conjugate Cartan subalgebras with spectrum $S^2$ in such crossed products. This is joint work with Johannes Ebert and Julian Kranz.[-]

Random walks is a topic at the crossroads of probability, ergodic theory, potential theory, harmonic analysis, geometry, and graph theory. Its roots can be traced back to the famous article by Pólya in 1921, which characterizes the recurrence of random walks on $\mathbb{Z}^{d}$ in terms of the dimension $d$. When random walks take place on a group, or more generally on a homogeneous space, it provides an even richer framework for study. From a probabilistic point of view, this additional structure serves as an extra tool, facilitating the study of the behaviour of the random walk on the underlying space. Regarding groups and their actions, random walks offer a means to explore generic or non-generic parts of groups and, at times, even to demonstrate intrinsic geometric properties, as is clearly shown by Kesten's amenability criterion (1959). This is an introductory course on the topic. Emphasis will be given on the interplay between probability and the structure of the group. The course will also provide insights into current research questions. Here is an outline of each session :
(1) Equivalent of Pólya's criterion for random walks on groups and rigidity theorems : does walking randomly on a given group in two different ways affect the recurrence of the walks ?
(2) Kesten's probabilistic criterion of the amenability of a finitely generated group ; defined in this course in terms of isoperimetric profile. The tools in 1) and 2) are essentially coming from analysis on groups.
(3) Tools coming from subadditivity to study the behaviour of a random walk on a group (drift, entropy and expansion of the random walk, etc.)[-]