|
Affine interval exchange transformations with flips and wandering intervals
Author(s):
C.
Gutierrez;
S.
Lloyd;
B.
Pires
Journal:
Proc. Amer. Math. Soc.
137
(2009),
1439-1445.
MSC (2000):
Primary 37E05, 37E10;
Secondary 37Bxx
Posted:
November 3, 2008
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 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)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
37E05, 37E10,
37Bxx
Retrieve articles in all Journals with MSC
(2000):
37E05, 37E10,
37Bxx
Additional Information:
C.
Gutierrez
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos - SP, Brazil
Email:
gutp@icmc.usp.br
S.
Lloyd
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australia
Email:
s.lloyd@unsw.edu.au
B.
Pires
Affiliation:
Departamento de Física e Matemática, Faculdade de Filosofia, Ciências e Letras da Universidade de São Paulo, Ribeirão Preto - SP, Brazil
Email:
benito@ffclrp.usp.br
DOI:
10.1090/S0002-9939-08-09718-9
PII:
S 0002-9939(08)09718-9
Received by editor(s):
February 28, 2008, and, revised form, May 31, 2008
Posted:
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
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org