In this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, i.e., the so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers (iwip) automorphisms) is given in the case where the dilation coefficient of the automorphism is a unit Pisot number. The shift map associated with the attractive symbolic lamination...

We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set.In the case of a compact surface, we get the following sharp result:...

The symbolic dynamical system associated with the Morse sequence is strictly ergodic. We describe some topological and metrical properties of the Cartesian powers of this system, and some of its other self-joinings. Among other things, we show that non generic points appear in the fourth power of the system, but not in lower powers. We exhibit various examples and counterexamples related to the property of weak disjointness of measure preserving dynamical systems.

It is shown that the set of learning systems having a singular stationary distribution is generic in the family of all systems satisfying the average contractivity condition.

We study Markov shifts over countable (finite or countably infinite) alphabets, i.e. shifts generated by incidence matrices. In particular, we derive necessary and sufficient conditions for the existence of a Gibbs state for a certain class of infinite Markov shifts. We further establish a characterization of the existence, uniqueness and ergodicity of invariant Gibbs states for this class of shifts. Our results generalize the well-known results for finitely irreducible Markov shifts.

We consider positional numeration systems with negative real base $-\beta$, where $\beta >1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $(-\beta )$-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base ${\beta }^2$ with a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy...

Let $𝒫$ be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to $𝒫$ is also in $𝒫$. Extending the classical Morse-Hedlund theorem, we show that either $𝒫$ contains at least $n+1$ words of length $n$ for every $n$ or, for some $N$, it contains at most $N$ words of length $n$ for every $n$. More importantly, we prove the following quantitative extension of this result: if $𝒫$ has $m\le n$ words of length $n$ then, for every $k\ge n+m$, it contains at most $\lceil (m+1)/2\rceil \lfloor (m+1)/2\rfloor$ words of length...

Let P be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to P is also in P. Extending the classical Morse-Hedlund theorem, we show that either P contains at least n+1 words of length n for every n or, for some N, it contains at most N words of length n for every n. More importantly, we prove the following quantitative extension of this result: if P has m ≤ n words of length n then, for every k ≥ n + m, it contains at most...

The Knaster continuum $K_p$ is defined as the inverse limit of the pth degree tent map. On every composant of the Knaster continuum we introduce an order and we consider some special points of the composant. These are used to describe the structure of the composants. We then prove that, for any integer p ≥ 2, all composants of $K_p$ having no endpoints are homeomorphic. This generalizes Bandt’s result which concerns the case p = 2.

In a 1982 paper Rauzy showed that the subshift $(X,T)$ generated by the morphism $1\mapsto 12$, $2\mapsto 13$ and $3\mapsto 1$ is a natural coding of a rotation on the two-dimensional torus ${𝕋}^2$, i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in ${ℝ}^2,$ each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity $2n+1$ satisfying a combinatorial criterion known as the $☆$ condition of Arnoux and Rauzy codes the orbit of a point...

Let (X,T) be a Cantor minimal system and let (R,) be the associated étale equivalence relation (the orbit equivalence relation). We show that for an arbitrary Cantor minimal system (Y,S) there exists a closed subset Z of X such that (Y,S) is conjugate to the subsystem (Z,T̃), where T̃ is the induced map on Z from T. We explore when we may choose Z to be a T-regular and/or a T-thin set, and we relate T-regularity of a set to R-étaleness. The latter concept plays an important role in the study of...

We consider endomorphisms of a monoid defined by a special confluent rewriting system that admit a continuous extension to the completion given by reduced infinite words, and study from a dynamical viewpoint the nature of their infinite periodic points. For prefix-convergent endomorphisms and expanding endomorphisms, we determine the structure of the set of all infinite periodic points in terms of adherence values, bound the periods and show that all regular periodic points are attractors.