Jan Boronski A class of compact minimal spaces whose Cartesian squares are not minimal

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1 Jan Boronski A class of compact minimal spaces whose Cartesian squares are not minimal Coauthors: Alex Clark and Piotr Oprocha In my talk I shall outline a construction of a family of 1-dimensional minimal spaces from [1], whose existence answer the following long standing problem in the negative. Problem. Is minimality preserved under Cartesian product in the class of compact spaces? Each space in the family is such that it admits a minimal homeomorphism but no Cartesian power of it does. Each such space also admits a monotone map onto a suspension of a minimal Cantor system, and the minimal homeomorphisms it admits are extensions of minimal homeomorphisms of the suspension. They are all factorwise rigidit, and their homeomorphisms groups are almost cyclic, in the sense that they are isomorphic to either $\mathbb{z}$ or $\mathbb{z}\otimes\mathbb{z}_2$. Note that for the fixed point property this question had been resolved in the negative already 50 years ago by Lopez [3], and a similar counterexample does not exist for flows, as shown by Dirbák [2]. References[1] Boronski J.P.; Clark A.; Oprocha P., A compact minimal space Y such that its square YxY is not minimal. arxiv: [2] Dirbák, M. Minimal extensions of flows with amenable acting groups. Israel J. Math. 207 (2015), no. 2, [3] Lopez, W. An example in the fixed point theory of polyhedra. Bull. Amer. Math. Soc Henk Don Slowly synchronizing automata with fixed alphabet size It was conjectured by Cerny in 1964 that a synchronizing deterministic finite automaton (DFA) on n states has a shortest synchronizing word of length at most (n-1)^2. This conjecture is still open and the best known bound is cubic in n. Examples for which the shortest synchronizing word attains the length (n-1)^2 are very sparse. Only one construction that works for general n is known, having alphabet size 2. In this talk I will explain the conjecture, discuss the role of the alphabet size, and give lower bounds for the maximal shortest synchronizing word length of a synchronizing DFA on n states and k symbols. Sebastien Ferenczi Rigidity for square tiled interval exchange transformations (joint work with Pascal Hubert) We start from a question of measure-theoretic dynamics, namely we want to find examples of nonrigid interval exchanges. Our answer comes from differentiable dynamics, as we consider first return maps on the set of diagonals of a flow of direction $\alpha$ on a square-tiled surface: our main result is to show that these are not rigid when the surface has at least one true singularity and $\alpha$ has bounded partial quotients. The proof uses symbolic dynamics with a word-combinatorics approach based on the scarcity of neighbours for the Hamming distance. Franz Gähler Towards an MLD classification of 1d inflation tilings 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.

2 It turns out that the many topological and dynamical invariants available, among them also some new ones, are quite powerful in distinguishing different MLD classes, even though a small number of unclear cases remains. It is quite possible, however, that the situation becomes much worse when going beyond ternary inflations. Uwe Grimm Substitution-based structures with absolutely continuous spectrum We derive new substitution-based structures with purely absolutely continuous diffraction and mixed dynamical spectrum, comprising absolutely continuous and pure point parts. This is achieved by generalising Rudin's construction of the binary Rudin-Shapiro sequence, and several examples are discussed in detail. An approach based on Fourier matrices yields constant-length substitutions for any length. This is joint work with Lax Chan and Ian Short. Henna Koivusalo From Diophantine approximation to cut and project sets Cut and project sets are, in many senses of the word, regular, but aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. Recently, a flexible formalism was discovered for translating information on Diophantine approximation to regularity properties of cut and project sets. In this talk I explain recent developments of the theory: how to quantify the relationship between Diophantine approximation and regularity properties of cut and project sets, and how to use this connection to gain detailed information on speed of convergence to asymptotics, of frequences of patterns in cut and project sets. The talk is based on joined work with Alan Haynes, Antoine Julien and Jamie Walton. Derong Kong Fractal properties in unique non-integer base expansions

3 Niels Langeveld N,alpha-continued fractions In this talk we look at a subfamily of the $N$-expansions. For this Family we find that the continued fraction expansions have digits from a finite alphabet. In most cases we are not able to find the invariant measure analytically but in some cases we are. In such case we show how to make the natural extension. For this subfamily of the $N$-expansions we can study the entropy as a function of $\alpha$. We show that for $N=2$ we have a plateaux on which the entropy function is constant. Stefano Marmi Diophantine type and dynamics of interval exchange maps and of translation flows The notion of diophantine type of an irrational number has equivalent characterizations in terms of rigidity of irrational rotations and linear flows on tori. Recently Roth type and higher type diophantine conditions have been introduced for translation flows on higher genus surfaces and interval exchange maps. Some rigidity results have also been proved, as well as some estimates on recurrence and hitting times, providing an extension of the theory beyond the torus case which is still incomplete but nevertheless quite broad. (Based on joint works with Moussa and Yoccoz and with Kim and Marchese). Milton Minervino Tree substitutions for Rauzy fractals Abstract: Rauzy fractals are well-known geometrical representations of Pisot substitutive dynamical systems. We show how to construct a self-similar tree which fills at the limit the Rauzy fractal. This gives dynamically interesting connections with interval exchange transformations. (Joint work with Thierry Coulbois). Hitoshi Nakada A construction of translation surfaces based on Cruz and da Rocha's idea for piecewise rotation maps of the circle Piotr Oprocha Invers limits of graphs and applications Combinatorial graphs can serve as a nice tool for description of dynamical systems on Cantor set. A classical example of this type are Bratelli- Vershik diagrams. Recently, Shimomura, motivated by works of Gambaudo and Martens, developed an alternative approach, which helps to describe dynamical systems on Cantor set by employing inverse limit of graphs. This approach provides a new useful tool for description of dynamical systems on Cantor set.

4 In this talk we will present fundamentals of this approach together with some applications (to some problems we were interested recently). Samuel Petite Restrictions on the automorphism group of a fixed subshift. A subshift is a closed set of sequences on a finite alphabet and invariant by the shift. An automorphism is an homeomorphism commuting with the shift map. Actually, it is a cellular automaton. It follows that the set of the automorphisms preserving a subshift is a countable group generally hard to describe. We will present in this talk a survey on different restrictions on these groups for zero entropy subshifts. Dan Rust Dynamics of Random Substitution Subshifts Random substitutions are a recent generalisation of the classical notion of a substitution, in which letters of the alphabet are mapped independently to a finite set of possible words rather than a single determined word. We are now beginning to understand the dynamical properties of subshifts associated to these random substitutions. I will give an overview of some recent developments stemming from joint work with Timo Spindeler. The question of how general these subshifts can be is a good one, and I will provide some results in this direction. I will outline a general construction linking random substitution subshifts to more traditional families of positive entropy subshifts. Wolfgang Steiner Recognizability for sequences of morphisms We investigate different notions of recognizability for a free monoid morphism $\sigma: A^* \to B^*$. Full recognizability occurs when each (aperiodic) two-sided sequence over $B$ admits at most one tiling with words $\sigma(a)$, $a \in A$. This is stronger than the classical notion of recognizability of a substitution $\sigma$, where the tiling must be compatible with the language of the substitution. We show that if $A$ is a two-letter alphabet, or if the incidence matrix of $\sigma$ has rank $ A $, or if $\sigma$ is permutative, then $\sigma$ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé (1992) and Bezuglyi, Kwiatkowski and Medynets (2009), by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define (eventual) recognizability for sequences of morphisms which define an $S$-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the $S$-adic shift it generates and the measurable Bratteli-Vershik dynamical system that it defines. This is joint work with Valérie Berthé, Jörg Thuswaldner and Reem Yassawi. Manon Stipulanti Generalized pascal triangle for binomial coefficients of finite words We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0,1]x[0,1] associated with this extended

5 Pascal triangle modulo a prime p. When p=2, we extract another sequence from a generalization of Pascal triangle in base 2 and study its regularity. Finally, we extend the results to the Fibonacci numeration system. Giulio Tiozzo On the local Hoelder exponent of the entropy function A question for your real analysis students: can you find a function whose local Hoelder exponent at any point equals the *value* of the function? It turns out that such functions arise quite naturally from dynamical systems, and we will see an example which comes from real unimodal maps. The proof is elementary and relies on the symbolic dynamics. James Walton Linear repetitivity of cut-and-project sets The two main sources of aperiodically ordered patterns are the cut-and-project method and tiling substitutions. Some questions are simple for patterns coming from one of these constructions but are difficult for those coming from the other. For example, it is easy to show that patterns coming from the cut-and-project method exhibit pure point diffraction, but the question is difficult for substitution tilings - the famous Pisot Conjecture on this remains unsolved. In the other direction, it is easy to show that all (primitive) substitution tilings are linearly repetitive, that is, there exists some C>0 for which every sub-patch of the pattern of size r occurs within radius Cr of any point of the pattern. I will discuss a recent result, generalising a classical result of Hedlund and Morse on Sturmian sequences, which states that a canonical codimension one cut-and-project set is linearly repetitive if and only if the physical space used in its construction corresponds to a badly approximable linear form.