Title: 1D substitutions as tilings and applications

Abstract:

In this talk I will present a graphical way to represent orbits of bi-infinite words under the action of a one-dimensional substitution. Then I will present two applications in symbolic dynamics on groups: one about surface groups and the other about amenable Baumslag-Solitar groups.

Based on a joint work with S. Barbierbi and E. Moutot and a joint work with M. Schraudner.

Title: Avoidability of Patterns

Abstract:

A pattern P is a string of variables. A string w avoids a pattern P, if for every substitution φ of the variables of P with non-empty words, φ(P) is not a factor of w. Thue initiated the study of avoidability by showing that there exists an infinite binary word that avoids pattern ABABA (overlaps), and an infinite ternary word that avoids pattern AA (squares). During the last century, this field of study has grown remarkably. This talk gives an overview of a selection of results concerning avoidability.

Title: Rauzy graphs, S-adicity and dendricity

Abstract:

Rauzy graphs are a classical and powerful tool to understand combinatorial properties of an infinite word such as its factor complexity or the frequencies of its factors. In this talk, I will present an overview of known results using them. I will then focus on the class of (eventually) dendric words.

Title: Infinite words connected to numeration: β-integers and Erdös spectrum

Abstract:

The famous Fibonacci word as the `nicest’ of sturmian sequences can be presented in a number of ways. Each of the definitions leads to family of infinite words that have the Fibonacci word as its special case. In the talk we will present two of the definitions that are connected to numeration systems. One of the possibilities is to see the Fibonacci word as a coding of distances between consecutive numbers with integer expansion in the numeration system having for base the golden ratio τ=(1+√5)/2, the so-called τ-integers. In general, β-integers with β being a Parry number provide an interesting family of pure morphic infinite words over a finite alphabet containing a class of sequences with affine factor complexity which do not belong to the category of interval exchange or Arnoux-Rauzy words. Another way to see the Fibonacci word in the frame of numeration systems is to consider the Erdös spectrum of τ. The spectrum of a real number β>1 with digit set *D*⊂*R* is defined as the set of polynomials with coefficients in *D* evaluated in β. If the base β is taken to be a Pisot number and the digits are consecutive integers including zero, the spectrum is a self-similar Delone set of finite local complexity which is coded by a morphic infinite word. We review the properties of such infinite words and show how geometric characteristics of the spectrum such as discreteness and relative density decide about good arithmetic behavior of the corresponding numeration system.

Title: Synchronized Sequences

Abstract:

The notion of synchronized sequence, introduced by Carpi and Maggi in 2001, has turned out to be a very useful tool for investigating the properties of words. In this paper I will prove some of the basic properties of synchronization, and give a number of applications to combinatorics on words.

**Luca Zamboni** (Université Claude Bernard Lyon 1, FR)

Title: Colouring problems in infinite words

Abstract:

Given a finite colouring (or finite partition) of the free semigroup *A*^{+} over a set *A*, we consider various types of monochromatic factorisations of one-sided infinite words *x*=*x*_{0}*x*_{1}*x*_{2}… in *A*^{ω}. We show how the existence of certain monochromatic factorisations can be used to characterise both periodicity and eventual periodicity in infinite words.

This is joint work with Maria-Romina Ivan and Imre Leader.