Mathematics – Dynamical Systems
Scientific paper
2010-05-11
Mathematics
Dynamical Systems
92 pages, 9 figures
Scientific paper
We study the dynamics of skew products of an irrational rotation with a discrete group. These are examples of infinite interval exchange maps which arise from (a minor modification to) Thurston's construction of pseudo-Anosov homeomorphisms of surfaces. Our version of Thurston's construction associates a one dimensional family of infinite interval exchange maps to a infinite connected bipartite graph G and a positive eigenfunction for the adjacency operator acting on functions on G. Dynamical properties of these infinite interval exchange maps including recurrence, topological conjugacies, and invariant measures are investigated. In many cases, the locally finite ergodic invariant measures are classified and shown to bijectively correspond to the extremal positive eigenfunctions of G. In the case of skew products of an irrational rotation with a nilpotent group, the locally finite ergodic invariant measures are the Maharam measures, in the cases when our classification theorem applies. In contrast, the locally finite ergodic invariant measures of some skew products of rotations and non-abelian free groups are classified in terms of the Gromov boundary of the group.
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