Samuel Petite

G-odometers and their almost one-to-one extensions

In this paper we recall the concepts of G-odometers and G-subodometers for G-actions, where G is a discrete finitely generated group; these generalize the notion of an odometer in the case G = . We characterize the G-regularly recurrent systems as the minimal almost one-to-one extensions of subodometers, from which we deduce that the family of the G-Toeplitz subshifts coincides with the family of the minimal symbolic almost one-to-one extensions of subodometers. We determine the continuous eigenvalues of these systems. When G is amenable and residually finite, a characterization of the G-invariant measures of these systems is given.

Invariant measures and orbit equivalence for generalized Toeplitz subshifts

We show that for every metrizable Choquet simplex K and for every group G, which is infinite, countable, amenable and residually finite, there exists a Toeplitz G-subshift whose set of shift-invariant probability measures is affine homeomorphic to K. Furthermore, we get that for every integer d > 1 and every Toeplitz flow (X, T ), there exists a Toeplitz Z-subshift which is topologically orbit equivalent to (X, T ).

Linearly repetitive delone sets

Linearly repetitive Delone sets are the simplest aperiodic repetitive Delone sets of the Euclidean space, e.g. any self similar Delone set is linearly repetitive.We present here some combinatorial, ergodic and mixing properties of their associated dynamical systems. We also give a characterization of such sets via the patch frequencies. Finally, we explain why a linearly repetitive Delone set is the image of a lattice by a bi-Lipschitz map.

Linearly repetitive Delone systems have a finite number of nonperiodic Delone system factors

-actions of the notion of subshift (see [Ro]). Classical ex-amples are those generated by self-similar tilings, as the Penrose one, whichhave been extensively studied since the 90’s. For details and references seefor example [Ro, So1]. Systems arising from self-similar tilings are knownto be linearly repetitive, this means there exists a positive constant L, suchthat every pattern of diameter D appears in every ball of radius LD inany tiling of the system. This concept has been first defined in [LP]. Lin-early repetitive tiling and Delone systems can be seen as a generalization toR

Eigenvalues and strong orbit equivalence

We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues E(X,T) of the minimal Cantor system (X,T) is a subgroup of the intersection I(X,T) of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated to (X,T) is trivial, the quotient group I(X,T)/E(X,T) is torsion free. We give examples with non trivial infinitesimal subgroups where this property fails. We also provide some realization results.

Self-Induced Systems

A minimal Cantor system is said to be self-induced whenever it is conjugate to one of its induced systems. Substitution subshifts and some odometers are classical examples, and we show that these are the only examples in the equicontinuous or expansive case. Nevertheless, we exhibit a zero entropy self-induced system that is neither equicontinuous nor expansive. We also provide non-uniquely ergodic self-induced systems with infinite entropy. Moreover, we give a characterization of self-induced minimal Cantor systems in terms of substitutions on finite or infinite alphabets.