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


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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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