Ergodic Theory and Dynamical Systems



On the dynamics of \mathbb{G}-solenoids. Applications to Delone sets


RICCARDO BENEDETTI a1 and JEAN-MARC GAMBAUDO a2
a1 Dipartimento di Matematica, Universitàa di Pisa, via F. Buonarroti 2, 56127 Pisa, Italy (e-mail: benedetti@dm.unipi.it)
a2 Institut de Mathématiques de Bourgogne, U.M.R. 5584 du CNRS, Université de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France (e-mail: gambaudo@u-bourgogne.fr)

Article author query
benedetti r   [Google Scholar] 
gambaudo j   [Google Scholar] 
 

Abstract

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

(Received September 2 2002)
(Revised January 22 2003)