2002/08/30 by Riccardo Benedetti, Benedetti, Riccardo, Jean-Marc Gambaudo +1 · 1 citation
Computer Science · Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #math.DS #msc:22F30 #msc:37C40 #msc:52C22
paper · pdf · doi:10.48550/arxiv.math/0208243
26 pages
arxiv created 2002/08/30 · arxiv updated 2009/11/30
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 Zd-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.