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Embeddings of interval exchange transformations into planar piecewise isometries

Part of: Ergodic theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  23 October 2018

PETER ASHWIN ,
AREK GOETZ ,
PEDRO PERES  and
ANA RODRIGUES
PETER ASHWIN
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK email [email protected]
AREK GOETZ
Affiliation:
Department of Mathematics, San Francisco State University, San Francisco, USA
PEDRO PERES
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK email [email protected]
ANA RODRIGUES
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK email [email protected]
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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.

Keywords

low-dimensional dynamicstopological dynamicspiecewise isometry

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 37E20: Universality, renormalization
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1153 - 1179
Copyright
© Cambridge University Press, 2018

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