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

TL;DR

This paper studies G-solenoids, a class of laminated dynamical systems with Lie group leaves, providing a topological framework and criteria for unique ergodicity, with applications to tiling spaces and Delone sets.

Contribution

It introduces a projective limit description of G-solenoids and characterizes their invariant measures using homology groups, offering new insights into their ergodic properties.

Findings

G-solenoids can be represented as projective limits of branched manifolds.

A topological description of invariant measures is provided via homology groups.

A simple criterion for unique ergodicity of G-solenoids is established.

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.