Symbolic Dynamics in connection with combinatorics on words, number theory and geometry.

Speaker : Charlene KALLE (Leiden University,Netherlands)

Ergodic theory is the field of mathematics that studies dynamical systems from a measure theoretic viewpoint; more precisely, it studies iteration of a map from a measurable set to itself which preserve a measure. Its theory connects to many field of mathematics, including probability, statistical mechanics, functional analysis and number theory, and one can use its tools to address questions in those fields. As a result of the measure theoretic approach, statements in ergodic theory often describe the typical behaviour of a system, that is, they hold for almost all points in the state space (this is a generalisation of the classical laws of large numbers in probability). The question of what happens on the exceptional set then creates a natural link to the field of fractal geometry as many of these sets are fractals. The course will introduce the basic concepts (measurable dynamical system, ergodicity, mixing...) and the main results (Poincaré recurrence theorem, ergodic theorems); it will also give a number of examples, focusing on number theory and its links to geometry.

Speaker : Karma DAJANI (University of Utrecht,Netherlands)

The course will introduce the notion of topological dynamical systems; more precisely, it studies the iteration of a continuous map from a topological space to itself; as such, is is the topological parallel to the previous course, and it will be taught along it. It will introduce the main important concepts (periodicity, recurrence, dense orbits, topological mixing, minimality) and basic theorems on recurrence and minimality, with various types of examples from geometry or combinatorics.

Speaker : Shigeki AKIYAMA (University of Tsukuba,Japan)

The course is intended, in part, to give a number of application to the two previous ones, will start later, and the last three lectures will take place on the last two days, to use the concepts and results of topological dynamics and ergodic theory. We start with a basic definition of symbolic dynamics and its applications to one-dimensional tilings, then extend the target to the 2 dimensional tiling problem including aperiodicity. 1) subshifts, sliding block code, subshift of finite type, sofic shifts, and graph representation related to them. The stress is on the coding of many dynamical systems. 2) Actual applications of symbolic dynamics, with an application to the important idea of Markoff partition. 3) Tiling could be understood as a higher dimensional version of symbolic dynamics. We discuss its periodicity, non-periodicity and the basic construction of non-periodic tilings by self-similarity. 4) Aperiodic Tile sets; a very interesting object emerges in 2 dimension. A finite set of tiles is called aperiodic if it tiles the space but only in non-periodic way. We discuss how the proof of aperiodicity works by interesting examples due to Ammann and Smith.

During this sessions, students will work in group (4/5 students) on open research questions presented by the teaching team. This activity will be coordinated by Pierre Arnoux ,the external organizer, who will also coordinate the programming sessions. We want to have a flexible format for this activity, which can also extend to some of the speed sessions and exercise sessions; it will also involve the other members of the teaching team, and it will be an occasion to use the programming activities.

Speaker : Eden Delight MIRO (Ateneo de Manila University,Philippines)

The first part of the course gives the basic elements of combinatorics on words (which will be used by all the advanced courses), and particularly the definition of factor complexity, and some classical examples of systems of low complexity. The second part will then focus on the main properties of substitution dynamical systems and their geometric models, and the basic examples, and will link with the course on tilings in the first week. The last part of the course will introduce the recent theory of random substitutions, the associated dynamical system, and the known results.

Speaker : Michel WALDSCHMIDT (Sorbonne University,France)

The goal of this course is to give a number of results, some (very) old, some very recent, linking arithmetic and diophantine properties of numbers to dynamical systems and combinatoric on words. The first part will review some classical number systems and the related dynamical systems (expansion in base $\beta$ and multiplication by $\beta$ mod 1, Continued fraction and Gauss map), and the elementary properties read from these expansions (rationality, Lagrange theorem, normal numbers). The second part will give classical diophantine approximation results : Dirichlet's box principle. Rational approximation to a real number: asymptotic and uniform. Dirichlet's Theorem, Liouville Theorem, and Thue--Siegel--Roth Theorem, and briefly expose the modern generalisations (Ridout Theorem, Schmidt's subspace theorem). The third part will explain recent examples of transcendental numbers, using expansions of low complexity statement, such as the fixed points of well-chosen substitutions.

Speaker : Renaud LEPLAIDEUR (Université de Nouvelle Calédonie,France)

The course will present substitutions on trees, one of the possible extensions of the classical notion of substitutions, and the results presently known on the recent field. If time allows, it will also present higher dimensional substitutions, and the applications to the theory of tilings.

The two programming sessions will introduce the participants to the SAGE software and show how it can be used in research.

Speaker : Renaud LEPLAIDEUR (Université de Nouvelle Calédonie,France)

Short research talks on their research by the participants