Conjugacy of unimodular Pisot substitutions subshifts to domain exchanges

We prove that any unimodular Pisot substitution subshift is measurably conjugate to a domain exchange in Euclidean spaces which factorizes onto a minimal rotation on a torus. This generalizes the pioneer works of Rauzy and Arnoux-Ito providing geometric realizations to any unimodular Pisot substitution without any additional combinatorial condition.Comment: 29 p. In this new version, a gap in the proof of the main theorem has been fixe

C*-algebras of Penrose's hyperbolic tilings

Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the C*-algebras and of the K-theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these C*-algebras are traceless. Nevertheless, harmonic currents give rise to 3-cyclic cocycles and we discuss in this setting a higher-order version of the gap-labelling.Comment: 36 pages. v2: some mistakes corrected, a section on topological invariants of the continuous hull of the Penrose hyperbolic tilings adde

On the simplicity of homeomorphism groups of a tilable lamination

We show that the identity component of the group of homeomorphisms that preserve all leaves of a R^d-tilable lamination is simple. Moreover, in the one dimensional case, we show that this group is uniformly perfect. We obtain a similar result for a dense subgroup of homeomorphisms.Comment: 14

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.Comment: 18

On automorphism groups of Toeplitz subshifts

In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. In the subquadratic complexity case, we show that the automorphism group modulo the torsion is generated by the roots of the shift map and that the result of the non superlinear case is optimal. Namely, for any $\varepsilon > 0$ we construct examples of minimal Toeplitz subshifts with complexity bounded by $C n^{1+\epsilon}$ whose automorphism groups are not finitely generated. Finally, we observe the coalescence and the automorphism group give no restriction on the complexity since we provide a family of coalescent Toeplitz subshifts with positive entropy such that their automorphism groups are arbitrary finitely generated infinite abelian groups with cyclic torsion subgroup (eventually restricted to powers of the shift)

Classification of discrete weak KAM solutions on linearly repetitive quasi-periodic sets

In discrete schemes, weak KAM solutions may be interpreted as approximations of correctors for some Hamilton-Jacobi equations in the periodic setting. It is known that correctors may not exist in the almost periodic setting. We show the existence of discrete weak KAM solutions for non-degenerate and weakly twist interactions in general. Furthermore, assuming equivariance with respect to a linearly repetitive quasi-periodic set, we completely classify all possible types of weak KAM solutions.Comment: 44 pages, 1 figur

Strong Approximations of Shifts and the Characteristic Measures Problem

Every symbolic system supports a Borel measure that is invariant under the shift, but it is not known if every such systems supports a measure that is invariant under all of its automorphisms; known as a characteristic measure. We give sufficient conditions to find a characteristic measure, additionally showing when it can be taken to be a measure of maximal entropy. The class of systems to which these sufficient conditions apply is large, containing a dense $G_{\delta}$ set in the space of all shifts on a given alphabet, and is also large in the sense that it is closed under taking factors. We also investigate natural systems to which these sufficient conditions apply