Michel Coornaert

Horofunctions and symbolic dynamics on Gromov hyperbolic groups

Let X be a proper geodesic metric space which is \delta -hyperbolic in the sense of Gromov. We study a class of functions on X , called horofunctions, which generalize Busemann functions. To each horofunction is associated a point in the boundary at infinity of X . Horofunctions are used to give a description of the boundary. In the case where X is the Cayley graph of a hyperbolic group \Gamma , we show, following ideas of Gromov sketched in his paper Hyperbolic groups , that the space of cocycles associated to horofunctions which take integral values on the vertices is a one-sided subshift of finite type.

A generalization of the Curtis-Hedlund theorem

Let A be a set and let G be a group, and equip A^G with its prodiscrete uniform structure. Let @t:A^G->A^G be a map. We prove that @t is a cellular automaton if and only if @t is uniformly continuous and G-equivariant. We also give an example showing that a continuous and G-equivariant map @t:A^G->A^G may fail to be a cellular automaton when the alphabet set A is infinite.

Topological dimension and dynamical systems

Translated from the popular French edition, the goal of the book is to provide a self-contained introduction to mean topological dimension, an invariant of dynamical systems introduced in 1999 by Misha Gromov. The book examines how this invariant was successfully used by Elon Lindenstrauss and Benjamin Weiss to answer a long-standing open question about embeddings of minimal dynamical systems into shifts. A large number of revisions and additions have been made to the original text. Chapter 5 contains an entirely new section devoted to the Sorgenfrey line. Two chapters have also been added: Chapter 9 on amenable groups and Chapter 10 on mean topological dimension for continuous actions of countable amenable groups. These new chapters contain material that have never before appeared in textbook form. The chapter on amenable groups is based on Folner s characterization of amenability and may be read independently from the rest of the book. Although the contents of this book lead directly to several active areas of current research in mathematics and mathematical physics, the prerequisites needed for reading it remain modest; essentially some familiarities with undergraduate point-set topology and, in order to access the final two chapters, some acquaintance with basic notions in group theory. Topological Dimension and Dynamical Systems is intended for graduate students, as well as researchers interested in topology and dynamical systems. Some of the topics treated in the book directly lead to research areas that remain to be explored

On the density of periodic configurations in strongly irreducible subshifts

Let G be a residually finite group and let A be a finite set. We prove that if X AG is a strongly irreducible subshift of finite type containing a periodic configuration then periodic configurations are dense in X. The density of periodic configurations implies in particular that every injective endomorphism of X is surjective and that the group of automorphisms of X is residually finite. We also introduce a class of subshifts , including all strongly irreducible subshifts and all irreducible sofic subshifts, in which periodic configurations are dense.

Groups, graphs, languages, automata, games and second-order monadic logic

In this paper we survey some surprising connections between group theory, the theory of automata and formal languages, the theory of ends, infinite games of perfect information, and monadic second-order logic.

Cellular automata on regular rooted trees

We study cellular automata on regular rooted trees. This includes the characterization of sofic tree shifts in terms of unrestricted Rabin automata and the decidability of the surjectivity problem for cellular automata between sofic tree shifts.

On algebraic cellular automata

We investigate some general properties of algebraic cellular automata, i.e., cellular automata over groups whose alphabets are affine algebraic sets and which are locally defined by regular maps. When the ground field is assumed to be uncountable and algebraically closed, we prove that such cellular automata always have a closed image with respect to the prodiscrete topology on the space of configurations and that they are reversible as soon as they are bijective.

AN UPPER BOUND FOR THE GROWTH OF CONJUGACY CLASSES IN TORSION-FREE WORD HYPERBOLIC GROUPS

We give a new upper bound for the growth of primitive conjugacy classes in torsion-free word hyperbolic groups.

Surjunctivity and Reversibility of Cellular Automata over Concrete Categories

Following ideas developed by Misha Gromov, we investigate surjunctivity and reversibility properties of cellular automata defined over certain concrete categories.

Cellular automata between sofic tree shifts

We study the sofic tree shifts of A^@S^^^@?, where @S^@? is a regular rooted tree of finite rank. In particular, we give their characterization in terms of unrestricted Rabin automata. We show that if X@?A^@S^^^@? is a sofic tree shift, then the configurations in X whose orbit under the shift action is finite are dense in X, and, as a consequence of this, we deduce that every injective cellular automata @t:X->X is surjective. Moreover, a characterization of sofic tree shifts in terms of general Rabin automata is given. We present an algorithm for establishing whether two unrestricted Rabin automata accept the same sofic tree shift or not. This allows us to prove the decidability of the surjectivity problem for cellular automata between sofic tree shifts. We also prove the decidability of the injectivity problem for cellular automata defined on a tree shift of finite type.