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)

Request Permissions   Purchase Content 


Rational mode locking for homeomorphisms of the $ 2$-torus

Authors: Salvador Addas-Zanata and Patrice Le Calvez
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 37E30, 37E45
Published electronically: December 26, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider homeomorphisms of the torus $ \mathbb{R}^2/\mathbb{Z}^2$, homotopic to the identity, and their rotation sets. Let $ f$ be such a homeomorphism, $ \widetilde {f}:\mathbb{R}^2\to \mathbb{R}^2$ be a fixed lift and $ \rho (\widetilde {f})\subset \mathbb{R}^2$ be its rotation set, which we assume to have interior. We also assume that the frontier of $ \rho (\widetilde {f})$ contains a rational vector $ \rho \in \mathbb{Q}^2$ and we want to understand how stable this situation is. To be more precise, we want to know if it is possible to find two different homeomorphisms $ f_1$ and $ f_2$, arbitrarily small $ C^0$-perturbations of $ f$, in a way that $ \rho $ does not belong to the rotation set of $ \widetilde f_1$ but belongs to the interior of the rotation set of $ \widetilde f_2,$ where $ \widetilde f_1$ and $ \widetilde f_2$ are the lifts of $ f_1$ and $ f_2$ that are close to $ \widetilde f$. We give two examples where this happens, supposing $ \rho =(0,0)$. The first one is a smooth diffeomorphism with a unique fixed point lifted to a fixed point of $ \widetilde f$. The second one is an area preserving version of the first one, but in this conservative setting we only obtain a $ C^0$ example. We also present two theorems in the opposite direction. The first one says that if $ f$ is area preserving and analytic, we cannot find $ f_1$ and $ f_2$ as above. The second result, more technical, implies that the same statement holds if $ f$ belongs to a generic one parameter family $ (f_t)_{t\in [0,1]}$ of $ C^2$-diffeomorphisms of $ \mathbb{T}^2$ (in the sense of Brunovsky). In particular, lifting our family to a family $ (\widetilde f_t)_{t\in [0,1]}$ of plane diffeomorphisms, one deduces that if there exists a rational vector $ \rho $ and a parameter $ t_*\in (0,1)$ such that $ \rho (\widetilde {f}_{{t_*}})$ has non-empty interior, and $ \rho \not \in \rho (\widetilde {f}_t)$ for $ t<t_*$ close to $ t_*$, then $ \rho \not \in \mathrm {int}(\rho (\widetilde {f}_{t}))$ for all $ t>t_*$ close to $ t_*$. This kind of result reveals some sort of local stability of the rotation set near rational vectors of its boundary.

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

  • [1] Salvador Addas-Zanata and Clodoaldo Grotta-Ragazzo, On the stability of some periodic orbits of a new type for twist maps, Nonlinearity 15 (2002), no. 5, 1385-1397. MR 1925419,
  • [2] Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5-42. MR 972342
  • [3] Philip Boyland, André de Carvalho, and Toby Hall, New rotation sets in a family of torus homeomorphisms, Invent. Math. 204 (2016), no. 3, 895-937. MR 3502068,
  • [4] Pavol Brunovský, On one-parameter families of diffeomorphisms, Comment. Math. Univ. Carolinae 11 (1970), 559-582. MR 0279827
  • [5] Freddy Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), no. 1, 53-106. MR 0650816,
  • [6] Jack Carr, Applications of centre manifold theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR 635782
  • [7] Freddy Dumortier, Paulo R. Rodrigues, and Robert Roussarie, Germs of diffeomorphisms in the plane, Lecture Notes in Mathematics, vol. 902, Springer-Verlag, Berlin-New York, 1981. MR 653474
  • [8] John Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), no. 1, 107-115. MR 958891,
  • [9] John Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8$ ^*$ (1988), no. Charles Conley Memorial Issue, 99-107. MR 967632,
  • [10] Jaume Llibre and Radu Saghin, The index of singularities of vector fields and finite jets, J. Differential Equations 251 (2011), no. 10, 2822-2832. MR 2831715,
  • [11] John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
  • [12] Michał Misiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490-506. MR 1053617,
  • [13] Michał Misiurewicz and Krystyna Ziemian, Rotation sets and ergodic measures for torus homeomorphisms, Fund. Math. 137 (1991), no. 1, 45-52. MR 1100607
  • [14] Jürgen K. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc. No. 81, Amer. Math. Soc., Providence, R.I., 1968. MR 0230498
  • [15] Floris Takens, Forced oscillations and bifurcations, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973) Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1-59. Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974. MR 0478235

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

Salvador Addas-Zanata
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil

Patrice Le Calvez
Affiliation: Sorbonne Universités, UPMC Univ Paris 06, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Univ. Paris Diderot, Sorbonne Paris Cité, F-75005, Paris, France

Received by editor(s): October 20, 2016
Received by editor(s) in revised form: March 3, 2017
Published electronically: December 26, 2017
Additional Notes: The first author was partially supported by CNPq grant 306348/2015-2
The second author was partially supported by CAPES, Ciencia Sem Fronteiras, 160/2012
Dedicated: This paper is dedicated to the memory of Lauro Antonio Zanata
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society