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Existence of non-trivial embeddings of interval exchange transformations into piecewise isometries

Part of: Ergodic theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  24 February 2025

PEDRO PERES  and
ANA RODRIGUES
PEDRO PERES
Affiliation:
University of Exeter, College of Engineering, Mathematics and Physics, Exeter EX4 4QE, UK (e-mail: [email protected])
ANA RODRIGUES*
Affiliation:
Departamento de Matemática, Escola de Ciências e Tecnologia, Universidade de Évora, Rua Romão Ramalho, 59, 7000–671 Évora, Portugal Centro de Investigação em Matemática e Aplicações, Rua Romão Ramalho, 59, 7000–671 Évora, Portugal
*
e-mail: [email protected]
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Abstract

We prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$, can be non-trivially and isometrically embedded in a family of piecewise isometries. In particular, this proves the existence of invariant curves for piecewise isometries, reminiscent of Kolmogorov–Arnold–Moser (KAM) curves for area-preserving maps, which are not unions of circle arcs or line segments.

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 , First View , pp. 1 - 39
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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