Course 2: Symbolic dynamics and aperiodic tiling

Instructor: Karma DAJANI

Course Description:

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

Course 3: Topological dynamical systems

Instructor: Shigeki AKIYAMA

Course Description:

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.

Course 4: Random substitutions

Instructor: Eden Delight MIRO

Course Description:

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.

Course 5: Diophantine approximation

Instructor: Michel WALDSCHMIDT

Course Description:

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.