Embeddings of interval exchange transformations into planar piecewise isometries
Ashwin, P; Goetz, A; Peres, P; et al.Rodrigues, A
Date: 23 October 2018
Journal
Ergodic Theory and Dynamical Systems
Publisher
Cambridge University Press (CUP)
Publisher DOI
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. ...
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 apparent abundance
of invariant nonsmooth fractal curves supporting IETs, that limit to a trivial embedding of
an IET.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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