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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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References

Adler, R., Kitchens, B., Martens, M., Tresser, C. and Wu, C. W.. The mathematics of halftoning. IBM J. Res. Dev. 47 (2003), 5–15.Google Scholar
Adler, R., Kitchens, B. and Tresser, C.. Dynamics of non-ergodic piecewise affine maps of the torus. Ergod. Th. & Dynam. Sys. 21 (2001), 959–999.Google Scholar
Arnoux, P., Orstein, D. and Weiss, B.. Cutting and stacking, interval exchanges and geometric models. Israel J. Math. 11 (1985), 160–168.Google Scholar
Ashwin, P.. Non-smooth invariant circles in digital overflow oscillations. Proceedings of the 4th Int. Workshop on Nonlinear Dynamics of Electronic Systems (Sevilla, 1996) pp. 417–422.Google Scholar
Ashwin, P., Chambers, W. and Petrov, G.. Lossless digital filter overflow oscillations; approximation of invariant fractals. Internat. J. Bifur. Chaos Appl. Engrg 7(11) (1997), 2603–2610.Google Scholar
Ashwin, P. and Fu, X.-C.. On the geometry of orientation-preserving planar piecewise isometries. J. Nonlinear Sci. 12(3) (2002), 207–240.Google Scholar
Ashwin, P. and Goetz, A.. Invariant curves and explosion of periodic islands in systems of piecewise rotations. SIAM J. Appl. Dyn. Syst. 4(2) (2005), 437–458.Google Scholar
Ashwin, P. and Goetz, A.. Polygonal invariant curves for a planar piecewise isometry. Trans. Amer. Math. Soc. 358(1) (2006), 373–390.Google Scholar
Avila, A. and Forni, G.. Weak mixing for interval exchange transformations and translation flows. Ann. of Math. (2) 165(2) (2007), 637–664.Google Scholar
Boshernitzan, M.. A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52 (1985), 723–752.Google Scholar
Bruin, H., Lambert, A., Poggiaspalla, G. and Vaienti, S.. Numerical analysis for a discontinuous rotation of the torus. Chaos 13(2) (2003), 558–571.Google Scholar
Buzzi, J.. Piecewise isometries have zero topological entropy. Ergod. Th. & Dynam. Sys. 5 (2001), 1371–1377.Google Scholar
Cornfeld, I. P., Fomin, S. V. and Sinai, Y. G.. Ergodic Theory (Grundlehren der Mathematisches Wissenschaften, 245). Springer, New York, 1982.Google Scholar
Davies, A. C.. Nonlinear oscillations and chaos from digital filters overflow. Phil. Trans. R. Soc. Lond. A 353 (1995), 85–99.Google Scholar
Deane, J. H. B.. Piecewise isometries: applications in engineering. Meccanica 41(3) (2006), 241–252. 37E99.Google Scholar
Galperin, G.. Two constructive sufficient conditions for aperiodicity of interval exchange. Theor. Appl. Prob. Opt. 176 (1985), 8–16.Google Scholar
Goetz, A.. Dynamics of piecewise isometries. PhD Thesis, University of Illinois at Chicago, 1996.Google Scholar
Goetz, A.. Dynamics of piecewise isometries. Illinois J. Math. 44 (2000), 465–478.Google Scholar
Haller, H.. Rectangle exchange transformations. Monatsh. Math. 91(3) (1981), 215–232.Google Scholar
Katok, A. B.. Interval exchange transformations and some special flows are not mixing. Israel J. Math. 35 (1980), 301–310.Google Scholar
Keane, M. S.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.Google Scholar
Kocarev, L., Wu, C. W. and Chua, L. O.. Complex behaviour in digital filters with overflow nonlinearity: analytical results. IEEE Trans. Circuits Systems II 43(3) (1996), 234–246.Google Scholar
Lowenstein, J. H. and Vivaldi, F.. Renormalization of a one-parameter family of piecewise isometries. Dynam. Syst. 31(4) (2016), 393–465.Google Scholar
Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115 (1982), 169–200.Google Scholar
Schwartz, R. E.. Unbounded orbits for outer billiards. J. Mod. Dyn. 3 (2007), 371–424.Google Scholar
Scott, A. J.. Hamiltonian mappings and circle packing phase spaces: numerical investigations. Phys. D 181 (2003), 45–52.Google Scholar
Scott, A. J., Holmes, C. A. and Milburn, G.. Hamiltonian mappings and circle packing phase spaces. Phys. D 155 (2001), 34–50.Google Scholar
Veech, W. A.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242.Google Scholar
Viana, M.. Ergodic Theory of interval exchange maps. Rev. Mat. Complut. 19 (2006), 7–100.Google Scholar