On the Dynamics of G-Solenoids. Applications to Delone Sets

Abstract

A G-solenoid is a laminated space whose leaves are copies of a single Lie group G, and whose transversals are totally disconnected sets. It inherits a G-action and can be considered as dynamical system. Free Z^d-actions on the Cantor set as well as a large class of tiling spaces possess such a structure of G-solenoid. We show that a G-solenoid can be seen as a projective limit of branched manifolds modeled on G. This allows us to give a topological description of the transverse invariant measures associated with a G-solenoid in terms of a positive cone in the projective limit of the dim(G)-homology groups of these branched manifolds. In particular we exhibit a simple criterion implying unique ergodicity. A particular attention is paid to the case when the Lie group $G$ is the group of affine orientation preserving isometries of the Euclidean space or its subgroup of translations.

26 pages

References in corpus (4)

• Tiling Spaces are Inverse Limits
• Spaces of tilings, finite telescopic approximations and gap-labelling
• Tiling spaces are Cantor set fiber bundles
• Repetitive Delone Sets and Quasicrystals

Cited by in corpus (9)

• Tiling Spaces are Inverse Limits
• Limit theorems for self-similar tilings
• Minimal configurations for Frenkel-Kontorova model on a quasicrystal
• Voronoi tessellations for matchbox manifolds
• Hierarchy of graph matchbox manifolds
• C*-algebras of tilings with infinite rotational symmetry
• On the geometry of ground states and quasicrystals for lattice systems
• Codimension zero laminations are inverse limits
• Meyer sets, topological eigenvalues, and Cantor fiber bundles