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- Title
Embeddings of interval exchange transformations into planar piecewise isometries.
- Authors
ASHWIN, PETER; GOETZ, AREK; PERES, PEDRO; RODRIGUES, ANA
- Abstract
Although piecewise isometries (PWIs) are higher-dimensional generalizations of one-dimensional interval exchange transformations (IETs), their generic dynamical properties seem to be quite different. In this paper, we consider embeddings of IET dynamics into PWI with a view to better understanding their similarities and differences. We derive some necessary conditions for existence of such embeddings using combinatorial, topological and measure-theoretic properties of IETs. In particular, we prove that continuous embeddings of minimal 2-IETs into orientation-preserving PWIs are necessarily trivial and that any 3-PWI has at most one non-trivially continuously embedded minimal 3-IET with the same underlying permutation. Finally, we introduce a family of 4-PWIs, with an apparent abundance of invariant non-smooth fractal curves supporting IETs, that limit to a trivial embedding of an IET.
- Publication
Ergodic Theory & Dynamical Systems, 2020, Vol 40, Issue 5, p1153
- ISSN
0143-3857
- Publication type
Article
- DOI
10.1017/etds.2018.112