Ergod. Th. & Dynam. Sys. (2003), 23, 673–691 c 2003 Cambridge University Press DOI: 10.1017/S0143385702001578 Printed in the United Kingdom On the dynamics of G-solenoids. Applications to Delone sets RICCARDO BENEDETTI† and JEAN-MARC GAMBAUDO‡ † Dipartimento di Matematica, Universita` di Pisa, via F. Buonarroti 2, 56127 Pisa, Italy (e-mail:
[email protected]) ‡ Institut de Mathematiques´ de Bourgogne, U.M.R. 5584 du CNRS, Universite´ de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France (e-mail:
[email protected]) (Received 2 September 2002 and accepted in revised form 22 January 2003) Abstract.AG-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 a dynamical system. Free Zd -actions on the Cantor set as well as a large class of tiling spaces possess such a structure of G-solenoids. For a large class of Lie groups, 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. 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. 1.