Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion


Authors: Andres Koropecki and Fabio Armando Tal
Journal: Proc. Amer. Math. Soc. 142 (2014), 3483-3490
MSC (2010): Primary 37E30, 37E45
DOI: https://doi.org/10.1090/S0002-9939-2014-12062-4
Published electronically: June 19, 2014
MathSciNet review: 3238423
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a $ C^\infty $ area-preserving diffeomorphism of the two-dimensional torus which is Bernoulli (in particular, ergodic) with respect to Lebesgue measure, homotopic to the identity, and has a lift to the universal covering whose rotation set is $ \{(0,0)\}$, which in addition has the property that almost every orbit by the lifted dynamics is unbounded and accumulates in every direction of the circle at infinity.


References [Enhancements On Off] (What's this?)

  • [AGT11] S. Addas-Zanata, B. Garcia, and F. A. Tal, Dynamics of homeomorphisms of the torus homotopic to Dehn twists, Ergodic Theory Dynam. Systems, published electronically in 2013, DOI 10.1017/etds.2012.156.
  • [Ano88] D. V. Anosov, On the behavior of trajectories, in the Euclidean or Lobachevskiĭ plane, covering the trajectory of flows on closed surfaces. II, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 451-478, 670 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 449-474. MR 954292 (89i:58125)
  • [AZ05] D. V. Anosov and E. V. Zhuzhoma, Nonlocal asymptotic behavior of curves and leaves of laminations on universal coverings, Tr. Mat. Inst. Steklova 249 (2005), 239 (Russian, with English and Russian summaries); English transl., Proc. Steklov Inst. Math. 2 (249) (2005), 1-219. MR 2200607 (2008d:37031)
  • [Dáv11] P. Dávalos, On torus homeomorphisms whose rotation set is an interval, Math. Z. 275 (2013), no. 3-4, 1005-1045. MR 3127045
  • [GS79] R. E. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc. 255 (1979), 403-414. MR 542888 (80k:58031), https://doi.org/10.2307/1998183
  • [Kat79] A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2) 110 (1979), no. 3, 529-547. MR 554383 (81a:28015), https://doi.org/10.2307/1971237
  • [KT12a] A. Koropecki and F. A. Tal, Bounded and unbounded behavior for area-preserving
    rational pseudo-rotations
    , Proc. London Math. Soc., to appear.
  • [KT12b] -, Strictly toral dynamics, Invent. Math. 196 (2014), no. 2, 339-381. MR 3193751
  • [MZ89] Michał Misiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490-506. MR 1053617 (91f:58052), https://doi.org/10.1112/jlms/s2-40.3.490
  • [OU41] J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874-920. MR 0005803 (3,211b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37E30, 37E45

Retrieve articles in all journals with MSC (2010): 37E30, 37E45


Additional Information

Andres Koropecki
Affiliation: Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brazil
Email: ak@id.uff.br

Fabio Armando Tal
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil
Email: fabiotal@ime.usp.br

DOI: https://doi.org/10.1090/S0002-9939-2014-12062-4
Received by editor(s): July 11, 2012
Received by editor(s) in revised form: September 14, 2012, October 1, 2012, and October 11, 2012
Published electronically: June 19, 2014
Additional Notes: The first author was partially supported by CNPq-Brasil.
The second author was partially supported by FAPESP and CNPq-Brasil
Communicated by: Nimish Shah
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society