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Cone exchange transformations and boundedness of orbits

Published online by Cambridge University Press:  07 September 2009

PETER ASHWIN  and
AREK GOETZ
PETER ASHWIN
Affiliation:
Mathematics Research Institute, University of Exeter, Exeter EX4 4QF, UK (email: [email protected])
AREK GOETZ
Affiliation:
San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA (email: [email protected])
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Abstract

We introduce a class of two-dimensional piecewise isometries on the plane that we refer to as cone exchange transformations (CETs). These are generalizations of interval exchange transformations (IETs) to 2D unbounded domains. We show for a typical CET that boundedness of orbits is determined by ergodic properties of an associated IET and a quantity we refer to as the ‘flux at infinity’. In particular we show, under an assumption of unique ergodicity of the associated IET, that a positive flux at infinity implies unboundedness of almost all orbits outside some bounded region, while a negative flux at infinity implies boundedness of all orbits. We also discuss some examples of CETs for which the flux is zero and/or we do not have unique ergodicity of the associated IET; in these cases (which are of great interest from the point of view of applications such as dual billiards) it remains an outstanding problem to find computable necessary and sufficient conditions for boundedness of orbits.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1311 - 1330
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Ashwin, P. and Goetz, A.. Polygonal invariant curves for a planar piecewise isometry. Trans. Amer. Math. Soc. 348 (2006), 373–390.Google Scholar
[2]Ashwin, P. and Goetz, A.. Invariant curves and explosion of periodic islands for piecewise isometries. SIAM J. Appl. Dyn. Syst. 4 (2005), 437–458.CrossRefGoogle Scholar
[3]Ashwin, P., Fu, X.-C. and Deane, J.. Properties of the invariant disk packing in a model bandpass sigma–delta modulator. Internat. J. Bifur. Chaos 13 (2003), 631–641.CrossRefGoogle Scholar
[4]Avila, A. and Forni, G.. Weak mixing for interval exchange transformations and translation flows. Ann. of Math. (2) 165 (2007), 637–664.CrossRefGoogle Scholar
[5]Boshernitzan, M. and Goetz, A.. A dichotomy for a two parameter piecewise rotation. Ergod. Th. & Dynam. Sys. 23 (2003), 759–770.CrossRefGoogle Scholar
[6]Boshernitzan, M. and Kornfeld, I.. Interval translation mappings. Ergod. Th. & Dynam. Sys. 15 (1995), 821–831.CrossRefGoogle Scholar
[7]Deane, J. H. B.. Global attraction in the sigma–delta piecewise isometry. Dyn. Syst. 17 (2002), 377–388.CrossRefGoogle Scholar
[8]Feely, O.. Nonlinear dynamics of bandpass sigma–delta modulation. Proceedings of NDES (Dublin, 1995), pp. 33–36.CrossRefGoogle Scholar
[9]Goetz, A. and Poggiaspalla, G.. Rotations by π/7. Nonlinearity 17 (2004), 1787–1802.CrossRefGoogle Scholar
[10]Goetz, A. and Quas, A.. Global properties of a family of piecewise isometries. Ergod. Th. & Dynam. Sys. 29 (2009), 545–568.CrossRefGoogle Scholar
[11]Bruin, H.. Renormalization in a class of interval translation maps of d branches. Dyn. Syst. 22 (2007), 11–24.CrossRefGoogle Scholar
[12]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 2004.Google Scholar
[13]Masur, H.. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115 (1982), 169–200.CrossRefGoogle Scholar
[14]Oxtoby, J. C.. Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116–136.CrossRefGoogle Scholar
[15]Schwartz, R. E.. Unbounded orbits for outer billiards. J. Modern Dynam. 3 (2007), 371–424.CrossRefGoogle Scholar
[16]Trovati, M. and Ashwin, P.. Tangency properties of a pentagonal tiling generated by a piecewise isometry. Chaos 17 (2007), 043129.CrossRefGoogle ScholarPubMed
[17]Veech, M.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242.CrossRefGoogle Scholar
[18]Vivaldi, F.. Aperiodic orbits of piecewise rational rotations of convex polygons with recursive tiling. Dyn. Syst. 22 (2007), 25–63.Google Scholar