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Affine interval exchange transformations with flips and wandering intervals

Authors: C. Gutierrez, S. Lloyd and B. Pires
Journal: Proc. Amer. Math. Soc. 137 (2009), 1439-1445
MSC (2000): Primary 37E05, 37E10; Secondary 37Bxx
Published electronically: November 3, 2008
MathSciNet review: 2465670
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Abstract: There exist uniquely ergodic affine interval exchange transformations of [0,1] with flips which have wandering intervals and are such that the support of the invariant measure is a Cantor set.

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  • 1. D. Berry and B. D. Mestel. Wandering intervals for Lorenz maps with bounded nonlinearity. Bull. London Math. Soc. 23 (1991), 183-189. MR 1122907 (93c:58119)
  • 2. A. M. Blokh and M. Yu. Lyubich. Non-existence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems, II. The smooth case. Ergod. Th. and Dyn. Sys. 9 (1989), 751-758. MR 1036906 (91e:58101)
  • 3. X. Bressaud, P. Hubert and A. Maass. Persistence of wandering intervals in self-similar affine interval exchange transformations. Preprint, arXiv:math.DS/08012088 (2007).
  • 4. R. Camelier and C. Gutierrez. Affine interval exchange transformations with wandering intervals. Ergod. Th. and Dyn. Sys. 17 (1997), no. 6, 1315-1338. MR 1488320 (99e:58117)
  • 5. M. Cobo. Piece-wise affine maps conjugate to interval exchanges. Ergod. Th. and Dyn. Sys. 22 (2002), no. 2, 375-407. MR 1898797 (2003h:37003)
  • 6. A. Denjoy. Sur le courbes définies par les équations différentielles à la surface du tore. J. Math. Pure et Appl. 11 (1932), série 9, 333-375.
  • 7. F. R. Gantmacher. Applications of the Theory of Matrices. Interscience, New York (1959). MR 0107648 (21:6372b)
  • 8. J. Guckenheimer. Sensitive dependence to initial conditions for one-dimensional maps. Commun. Math. Phys. 70 (1979), 133-160. MR 553966 (82c:58037)
  • 9. C. Gutierrez, S. Lloyd, V. Medvedev, B. Pires and E. Zhuzhoma. Unique ergodicity of circle and interval exchange transformations with flips. Preprint, arXiv:math.DS/07113821 (2007).
  • 10. M. Keane. Non-ergodic interval exchange transformations. Israel J. Math. 26 (1977), no. 2, 188-196. MR 0435353 (55:8313)
  • 11. G. Levitt. La décomposition dynamique et la différentiabilité des feuilletages des surfaces. Ann. Inst. Fourier (Grenoble) 37 (1987), 85-116. MR 916275 (88m:57035)
  • 12. I. Liousse and H. Marzougui. Échanges d'intervalles affines conjugués à des linéaires. Ergod. Th. and Dyn. Sys. 22 (2002), no. 2, 535-554. MR 1898804 (2003c:37050)
  • 13. M. Martens, W. de Melo and S. van Strien. Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168 (1992), 273-318. MR 1161268 (93d:58137)
  • 14. H. Masur. Interval exchange transformations and measured foliations. Ann. of Math. (2) 115 (1982), 169-200. MR 644018 (83e:28012)
  • 15. A. Nogueira. Almost all interval exchange transformations with flips are nonergodic. Ergod. Th. and Dyn. Sys. 9 (1989), no. 3, 515-525. MR 1016669 (91d:28035)
  • 16. G. Rauzy. Échanges d'intervalles et transformations induites. Acta Arith. 34 (1979), 315-328. MR 543205 (82m:10076)
  • 17. W. Veech. Interval exchange transformations. J. d'Analyse Math. 33 (1978), 222-272. MR 516048 (80e:28034)
  • 18. W. Veech. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201-242. MR 644019 (83g:28036b)
  • 19. Y. C. Yoccoz. Il n'y a pas de contre-exemple de Denjoy analytique. C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 7, 141-144. MR 741080 (85j:58134)

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Additional Information

C. Gutierrez
Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos - SP, Brazil

S. Lloyd
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australia

B. Pires
Affiliation: Departamento de Física e Matemática, Faculdade de Filosofia, Ciências e Letras da Universidade de São Paulo, Ribeirão Preto - SP, Brazil

Received by editor(s): February 28, 2008
Received by editor(s) in revised form: May 31, 2008
Published electronically: November 3, 2008
Additional Notes: The first author was partially supported by FAPESP Grant 03/03107-9 and by CNPq Grants 470957/2006-9 and 306328/2006-2.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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