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.[-]

Dimension groups are invariants of orbital equivalence. We show in this lecture how to compute the dimension group of tree subshifts. Tree subshifts are defined in terms of extension graphs that describe the left and right extensions of factors of their languages: the extension graphs are trees. This class of subshifts includes classical families such as Sturmian, Arnoux-Rauzy subshifts, or else, codings of interval exchanges. We rely on return word properties for tree subshifts: every finite word in the language of a tree word admits exactly d return words, where d is the cardinality of the alphabet.
This is joint work with P. Cecchi, F. Dolce, F. Durand, J. Leroy, D. Perrin, S. Petite.[-]

Subshifts on finite alphabets form a class of dynamical systems that bridge topological/ergodic dynamical systems with that of word combinatorics. In 1984, M. Boshernitzan used word combinatorics to provide a bound on the number of ergodic measures for a minimal subshift with bounds on its linear factor complexity growth rate. He further asked if the correct bound for subshifts naturally coded by interval exchange transformations (IETs) could be obtained by word combinatoric methods. (The ”correct” bound is roughly half that obtained by Boshernitzan's work.) In 2017 and joint with M. Damron, we slightly improved Boshernitzan's bound by restricting to a smaller class of subshifts that still contained IET subshifts. In recent work, we have further proved the ”correct” bound to subshifts whose languages satisfy a specific word combinatoric condition, which we called the Regular Bispecial Condition. (This condition is equivalent to being Eventually Dendric as independently introduced by F. Dolce and D. Perrin.)
During the same time we worked on our 2017 paper, V. Cyr and B. Kra were independently improving Boshernitzan's results. In 2019, they relaxed the conditions to no longer require minimality and extended Boshernitzan's bound to generic measures. (Generic measures are those that have generic points, meaning they satisfy the averaging limits as stated in Pointwise Ergodic Theorem. However, there are non-ergodic generic measures.) We have obtained the improved 2017 bound but for generic measures (and on a more general class of subshifts). It should be noted that, to our current knowledge, there does not exist a proof of the correct bound of generic measures for minimal IETs (by any method).In this talk, I will discuss these recent results and highlight related open problems.[-]

We consider infinite interval exchange transformations (IETs) obtained as a composition of a finite IET and the von Neumann-Kakutani map, called rotated odometers, and study their dynamical and ergodic properties by means of an associated Bratteli-Vershik system. We show that every rotated odometer is measurably isomorphic to the first return map of a rational parallel flow on a translation surface of finite area with infinite genus and a finite number of ends, with respect to the Lebesgue measure. This is one motivation for the study of rotated odometers. We also prove a few results about the factors of the unique minimal subsystem of a rotated odometer. This is joint work with Henk Bruin.[-]

We study skew products of circle diffeomorphisms over a shift space. Our primary motivation is the fact that they capture some key mechanisms of nonhyperbolic behavior of robustly transitive dynamical systems. We perform a multifractal analysis of fiber-Lyapunov exponents studying the topological entropy of fibers with equal exponent. This includes the study of restricted variational principles of the entropy of ergodic measures with given fiber-exponent, in particular, with exponent zero. This enables to understand transitive dynamical systems in which hyperbolicities of different type are intermingled. Moreover, it enables to 'quantify of the amount of non-hyperbolicity' in a context where any other tools presently available fail. This is joint work with L.J. Díaz and M. Rams.[-]

An automorphism of a subshift $X$ is a self-homeomorphism of $X$ that commutes with the shift map. The study of these automorphisms started at the very beginning of the symbolic dynamics. For instance, the well known Curtis-Hedlund-Lyndon theorem asserts that each automorphism is a cellular automaton. The set of automorphisms forms a countable group that may be very complicated for mixing shift of finite type (SFT). The study of this group for low complexity subshifts has become very active in the last five years. Actually, for zero entropy subshift, this group is much more tame than in the SFT case. In a first lecture we will recall some striking property of this group for subshift of finite type. The second lecture is devoted to the description of this group for classical minimal sub shifts of zero entropy with sublinear complexity and for the family of Toeplitz subshifts. The last lecture concerns the algebraic properties of the automorphism group for subshifts with sub-exponential complexity. We will also explain why sonic group like the Baumslag-Solitar $BS(1,n)$ or $SL(d,Z), d >2$, can not embed into an automorphism group of a zero entropy subshift.[-]

Subshifts of finite type are of high interest from a computational point of view, since they can be described by a finite amount of information - a set of forbidden patterns that defines the subshift - and thus decidability and algorithmic questions can be addressed. Given an SFT $X$, the simplest question one can formulate is the following: does $X$ contain a configuration? This is the so-called domino problem, or emptiness problem: for a given finitely presented group $0$, is there an algorithm that determines if the group $G$ is tilable with a finite set of tiles? In this lecture I will start with a presentation of two different proofs of the undecidability of the domino problem on $Z^2$. Then we will discuss the case of finitely generated groups. Finally, the emptiness problem for general subshifts will be tackled.[-]

The theorem of Büchi-Bruyère states that a subset of $N^d$ is $b$-recognizable if and only if it is $b$-definable. As a corollary, the first-order theory of $(N,+,V_b)$ is decidable (where $V_b(n)$ is the largest power of the base $b$ dividing $n$). This classical result is a powerful tool in order to show that many properties of $b$-automatic sequences are decidable. The first part of my lecture will be devoted to presenting this result and its applications to $b$-automatic sequences. Then I will move to $b$-regular sequences, which can be viewed as a generalization of $b$-automatic sequences to integer-valued sequences. I will explain bow first-order logic can be used to show that many enumeration problems of $b$-automatic sequences give rise to corresponding $b$-regular sequences. Finally, I will consider more general frameworks than integer bases and (try to) give a state of the art of the research in this domain.[-]