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Almost all interval exchange transformations with flips are nonergodic†

Published online by Cambridge University Press:  19 September 2008

Arnaldo Nogueira
Arnaldo Nogueira
Affiliation:
Institute de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944 Rio de Janeiro, RJ, Brasil
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Abstract

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Here we prove that almost all interval exchange transformations which reverse orientation, in at least one interval, have a periodic point where the derivative is − 1. Therefore they are periodic in an open neighborhood of the periodic point.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 3 , September 1989 , pp. 515 - 525
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[1]Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.CrossRefGoogle Scholar
[2]Sataev, E.A.. On the number of invariant measures of flows on orientable surfaces. Izv. Acad. Sci. USSR, Ser. Math. 39(4) (1970), 860–878.Google Scholar
[3]Keane, M.. Non-ergodic interval exchange transformations. Israel J. Math. 26(2) (1977), 188–196.CrossRefGoogle Scholar
[4]Masur, H.. Interval exchange transformations and measured foliations. Ann. Math. 115 (1982), 169–200.CrossRefGoogle Scholar
[5]Veech, W.A.. Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 115 (1982), 201–242.CrossRefGoogle Scholar
[6]Veech, W.A.. Projective Swiss Cheeses and uniquely ergodic interval exchange transformations. Ergod. Th. & Dynam. Sys. 1, in: Progress in Mathematics, pp 113–193. Bierkhauser: Boston, 1981.Google Scholar
[7]Gutierrez, C.. Smooth nonorientable nontrivial recurrence on two-manifolds. J. Diff. Eq. 29(3) (1978), 388–395.CrossRefGoogle Scholar
[8]Nogueira, A.. Nonorientable recurrence of flows and interval exchange transformations. J. Diff. Equations. 70(2) (1987), 153–166.CrossRefGoogle Scholar
[9]Rauzy, G.. Eschanges d'intervalles et transformations induites. Acta Arith. 34 (1979), 315–328.CrossRefGoogle Scholar
[10]Veech, W.A.. Interval exchange transformations. J. d'Analyse Math. 33 (1978), 222–278.CrossRefGoogle Scholar
[11]Kerckhoff, S.P.. Simplicial systems for interval exchange maps and measured foliations. Ergod. Th.& Dynam. Sys. 5 (1985), 257–271.CrossRefGoogle Scholar