We introduce the notions of asymptotic period and asymptotically periodic orbits in metric spaces. We study some properties of these notions and their connections with ω-limit sets. We also discuss the notion of growth rate of such orbits and describe its properties in an extreme case.

The dynamical Mertens' theorem describes asymptotics for the growth in the number of closed orbits in a dynamical system. We construct families of ergodic automorphisms of fixed entropy on compact connected groups with a continuum of growth rates on two different growth scales. This shows in particular that the space of all ergodic algebraic dynamical systems modulo the equivalence of shared orbit-growth asymptotics is not countable. In contrast, for the equivalence relation of measurable isomorphism...

We consider the billiard map in the hypercube of ${ℝ}^d$. We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that $n^{3d-3}$ is the order of magnitude of the complexity.

Ces notes ont pour but de rassembler les différents résultats de combinatoire des mots relatifs au billard polygonal et polyédral. On commence par rappeler quelques notions de combinatoire, puis on définit le billard, les notions utiles en dynamique et le codage de l’application. On énonce alors les résultats connus en dimension deux puis trois.

We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the complexity has cubic asymptotic growth.

We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose...

In this paper we study dynamical properties of linear actions by free groups via the induced action on projective space. This point of view allows us to introduce techniques from Thermodynamic Formalism. In particular, we obtain estimates on the growth of orbits and their limiting distribution on projective space.

Given β > 1, let Tβ$\begin{array}{ccc}{\mathit{T}}_{\mathit{\beta }}\mathrm{:}\mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}& \mathrm{\to }& \mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}\\ \mathit{x}& \mathrm{\to }& \mathit{\beta x}\mathrm{-}\mathrm{\lfloor }\mathit{\beta x}\mathrm{\rfloor }\mathit{.}\end{array}$The iteration of this transformation gives rise to the greedy β-expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter β.In [17], K. Schmidt analyzed the set of periodic points of Tβ, where β is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansion$\begin{array}{ccc}\mathit{x}\mathrm{=}\sum _{\mathit{i}\mathrm{\ge }\mathrm{0}}{\mathit{e}}_{\mathit{i}}{\mathit{\beta }}^{\mathrm{-}\mathit{i}}\mathit{,}& & \end{array}$where each ei...

Nous précisons le comportement exponentiel de la fonction orbitale d'un quelconque groupe discret d'isométries en courbure négative.