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.[-]

The pure point or discrete spectrum is a fundamental dynamical property often linked to the invariant measures of group actions associated to quasicrystals The Halmos - von Neumann representation theory shows that Measure-preserving actions with pure point spectrum inherently possess a straightforward and uniquely ergodic topological model, namely group rotations. However, it's worth noting that every measure-preserving action can have more intricate topological models. For instance, one can derive topologically mixing models for actions exhibiting pure point spectrum. In this presentation, we aim to provide a comprehensive overview of recent
ndings regarding ALL conceivable topological models for pure point spectrum actions, with particular emphasis on the natural classes that emerge within this framework.[-]

The automorphism group of a symbolic system captures its symmetries, reecting the dynamical behavior and the complexity of the system. It can be quite complicated: for example, for a topologically mixing shift of
nite type, the automorphism group contains isomorphic copies of all
nite groups and the free group on two generators and such behavior is common for shifts of high complexity. In the opposite setting of low complexity, there are numerous restrictions on the automorphism group, and for many classes of symbolic systems, it is known to be virtually abelian. I will give an overview of relations among dynamical properties of the system, algebraic properties of the automorphism group, and measurable properties of associated systems, all of which quickly lead to open questions.[-]