**Speakers**

María Isabel Cortez (Universidad de Santiago de Chile): *Invariant measures of actions of amenable monotileable groups on the Cantor set.*

Basile de Loynes (ENSAI): *Random walks on graphs induced by aperiodic tilings. [Slides]*

Jordan Emme (Université Paris Sud): *A regularity property of spectral measures of self-similar tilings.*

Franz Gähler (Universität Bielefeld): *Towards an MLD classification of 1d inflation tilings.*

Guilhem Gamard (Higher School of Economics, Moscow): *Quasiperiodicity as a local rule. [Slides]*

Chaim Goodman-Strauss (University of Arkansas): *Strongly aperiodic subshifts of finite type on hyperbolic groups. [Slides]*

Uwe Grimm (The Open University): *Spectral Properties of Aperiodic Structures. [Slides]*

Martha Łącka (Jagiellonian University, Kraków): *The Weyl Pseudometric.*

Paul-Henry Leemann (ENS de Lyon): **Limit of Rauzy graphs of subshifts of finite type.**

Daniel Lenz (UniversitätJena): *An autocorrelation and discrete spectrum for dynamical systems on metric spaces.*

Nicolas Ollinger (Université d'Orléans): **Substitutions and Strongly Deterministic Tilesets. [Slides]**

Samuel Petite (Université de Picardie Jules Verne): **Non expansive directions for group actions.**

Andrei Romashchenko (LIRMM and CNRS): *Embedding computations in hierarchical tilings. [Slides]*

**Lorenzo Sadun** (University of Texas at Austin): *Homeomorphisms of tiling spaces. [Slides]*

Timo Spindeler (Universität Bielefeld): *Random substitutions and shifts of finite type.*

Michal Szabados (University of Turku) : *Pattern complexity on the edge between periodicity and aperiodicity. [Slides]*

Ilkka Törmä (University of Turku): *Multidimensional SFTs with countably many configurations. [Slides]*

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**Abstracts**

** **

María Isabel Cortez: *Invariant measures of actions of amenable monotileable groups on the Cantor set.*

*Abstract: **The set of invariant probability measures of a continuous action of an amenable group on a compact metric space is a (non empty) metrizable Choquet simplex. A natural question is to know if the converse is true, i.e, if given a metrizable Choquet simplex K and an amenable group G, it is possible to realize K as the set of invariant probability measures of a continuous action of G on a compact metric space. In 1991 Downarowicz answered for the first time this question in the case G = Z, showing that every metrizable Choquet simplex can be realized as the set of invariant probability measures of a Toeplitz Z-subshift. In 2014 it was shown that this is also true if G is any amenable residually finite group (C, Petite 2014). In this talk we show the extension of this result to some larger class of monotileable groups (Cecchi, C.), which include all the nilpotent groups (even those which are not finitely generated).*

*Basile de Loynes: **Random walks on graphs induced by aperiodic tilings.*

*Abstract: **T**he cut-and-project method is a geometric construction introduced by Oguey, Duneau and Katz (1988) in order to define aperiodic tilings which naturally induce graphs embedded in the Euclidean space. The aperiodic structure of the tilings implies the graphs are no longer homogeneous spaces. In this talk, two results related to the simple random walk will be presented. The first one is related the recurrent/transience type of the simple random walk. The second one gives an estimate of its asymptotic entropy. As these results are very similar to the usual case of simple random walk on the integer lattice, it suggests that a varying local curvature does not modify the global behavior of the stochastic process as long as the graph remains roughly globally flat.*

*Jordan Emme: **A regularity property of spectral measures of self-similar tilings.*

**Abstract:** In ergodic theory, many interesting dynamical properties of measured dynamical systems are of spectral nature. Spectral measures were introduced in order to understand the spectrum of the Koopman operator of a dynamical system so as to study these properties. A.Bufetov and B.Solomyak have proven a regularity property for self-similar suspension flows over substitutive dynamical systems. Their result is based on a previous theorem describing precisely ergodic deviations in the case of the translation flow of a self-similar tiling, which was proven with the formalism of finitely additive measures introduced by A.Bufetov. In this talk we give a generalisation of their result, stating the same sort of regularity for spectral measures associated to translation flows on self-similar tilings.

*Franz Gähler: **Towards an MLD classification of 1d inflation tilings.*

*Abstract: **We consider the family of all ternary, unimodular, irreducible **Pisot inflation tilings with an inflation factor among the eight*

smallest ternary Pisot units. We investigate to what extent we can distinguish or identify the MLD classes of these tilings, or more precisely their hulls and (translation) dynamical systems. Tilings are MLD (mutually locally derivable) if their dynamical

systems are topologically conjugate with a local conjugation map. It turns out that the many topological and dynamical invariants

available, among them also some new ones, are sufficiently powerful to distinguish the different MLD classes occurring. It is quite

possible, however, that the situation becomes much worse when going beyond ternary inflations.

Guilhem Gamard: *Quasiperiodicity as a local rule.*

*Abstract: **Quasiperiodicity is usually defined as a notion of regularity: a pattern q is a quasiperiod of coloring of ℤ², say w, if and only if w is covered with occurrences of q (which may overlap). First we'll survey known results about quasiperiodicity of colorings of ℤ². Then we'll change our point of view and see quasiperiodicity as a local rule. We'll try to understand what global properties may be forced by this particular local rule.*

Chaim Goodman-Strauss: *Strongly aperiodic subshifts of finite type on hyperbolic groups.*

**Abstract:** Every one-ended word-hyperbolic group admits a strongly aperiodic subshift of finite type. (Joint with David Cohen, U. Chicago, and Yo'av Rieck, U. Arkansas)

*Uwe Grimm: **Spectral Properties of Aperiodic Structures.*

*Abstract: **The characterisation of aperiodic structures usually employs spectral notions and properties. In particular, this applies to Fourier analysis in the form of diffraction measures, which are directly linked to experimental observations, for instance in the structure analysis of quasicrystalline materials. While diffraction provides information about the order in a system, we are still far from any classification of systems displaying some form of long-range order. Many of the aperiodic structures that are considered give rise, under translation action, to ergodic dynamical systems. There is then a natural spectral measure associated to this dynamical system, which is the dynamical spectrum. In my talk, I will discuss the relation between these two spectral notions and discuss some explicit examples.*

*Martha Łącka: **The Weyl Pseudometric.*

*Abstract: **During the talk we will present some consequences of the convergence with respect to the Weyl pseudometric in dynamical systems generated by an amenable group actions. This will lead us to the alternative proof of the Krieger theorem, which says that for any number between 0 and log k one can find a Toeplitz shift over a k-letter alphabet with entropy equal to this number.*

*Paul-Henry Leemann:*** Limit of Rauzy graphs of subshifts of finite type.**

**Abstract:** To each SFT on Z it is possible to associate an infinite family of finite graphs, the so called Rauzy graphs. For a Markovian subshift defined by a matrix $M$, these graphs are the one with adjacency matrices $M^n$. In this talk, we will compute the weak limit (also called local limit or Benjamini-Schramm) of these graphs and gave two description of it. One in term of the unique measure of maximal entropy and the other in terms of horospheric product of trees, which are related to the Lamplighter group. (Joint with T. Nagnibeda , U. Genève, and V. Kaimanovich, U. Ottawa)

Daniel Lenz: *An autocorrelation and discrete spectrum for dynamical systems on metric spaces.*

*Abstract: *We study dynamical systems (X,G,m) with a compact metric space X, a locally compact, σ -compact, abelian group G and an invariant probability measure m. We show that such a system has discrete spectrum if and only if a certain space average over the metric is a Bohr almost periodic function. In this way, this average over the metric plays for general dynamical systems a similar role as the autocorrelation measure plays in the study of aperiodic order for special dynamical systems based on point sets.

*Samuel Petite: ***Non expansive directions for group actions.**

**Abstract: **In their seminal work on symbolic dynamics, Morse and Hedlund show the existence of asymptotic pairs of sequences in any infinite subshift, by introducing the notion of special words. Later, this notion has been extended in multidimensional Z^d-subshift by Boyle and Lind, via the notion of non expansive direction, and to R^d-tilings spaces by Barge and Olimb. We propose an extension for non abelian group action and we give some consequences.

*Andrei Romashchenko: **Embedding computations in hierarchical tilings.*

**Abstract:** We discuss a generic construction of hierarchical SFT based on a technique of self-referential programming that goes back to Kleene's fixed-point theorem and von Neumann's self-reproducing automata. This construction allows to embed in the geometric structure of a subshift the space-time diagram of any computation. We will show how to use the embedded computation to enforce non-trivial combinatorial and topological properties of an SFT. As an illustration, we will start with the classic Berger's result on aperiodic tilings and proceed with the Hochman-Meyerovitch characterization of the entropies of SFTs, with additional properties of irreducibility and transitivity.

**Lorenzo Sadun**: *Homeomorphisms of tiling spaces.*

*Abstract: **We show that every homeomorphism between FLC tiling spaces is a product of (1) a local** (MLD) map, (2) a map homotopic to the identity, and (3) a shape change. Furthermore, the possible H*^{1}(Ω, R^{n}). Similar results apply to shape changes are parametrized by a dense open set of classes in *ILC tiling spaces. This is joint work with Antoine Julien.*

Timo Spindeler: *Random substitutions and shifts of finite type.*

**Abstract:** *Random substitutions are a natural generalisations of their classical 'deterministic' counterpart, whereby at every step of iterating the substitution, instead of replacing a letter with a predetermined word, every letter is independently replaced by a word from a finite set of possible words according to a probability distribution. Moreover, we establish connections to shifts of finite type.*

*Michal Szabados: **Pattern complexity on the edge between periodicity and aperiodicity.*

*Abstract: *A one-dimensional infinite word is either periodic, or the number of subwords of length n is at least n+1 for all n. This relation between subword complexity and periodicity was proved by Morse and Hedlund, and gives rise to definition of Sturmian words as aperiodic words with minimal complexity.

We will try to generalize the statement to two and more dimensions. We will study configurations of low complexity and show that while they might not necessarily be periodic, they decompose into a sum of periodic components. We will use polynomials and a bit of algebraic geometry to obtain the result.

Ilkka Törmä: *Multidimensional SFTs with countably many configurations.*

**Abstract:** A recurring theme in the theory of tilings is the construction of tile sets whose associated shift spaces are in some sense as restricted as possible. Aperiodic SFTs are one example: all configurations are constrained to have no global period. We consider another restriction, namely, that the set of valid configurations has countable cardinality. Such tile sets cannot be aperiodic, but the geometry of their configurations is not as constrained as one might think. In particular, it is possible to carry out certain hierarchical constructions.