Rational mode locking for homeomorphisms of the $2$-torus
HTML articles powered by AMS MathViewer
- by Salvador Addas-Zanata and Patrice Le Calvez PDF
- Proc. Amer. Math. Soc. 146 (2018), 1551-1570 Request permission
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
- 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, DOI 10.1088/0951-7715/15/5/302
- Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342
- 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, DOI 10.1007/s00222-015-0628-2
- Pavol Brunovský, On one-parameter families of diffeomorphisms, Comment. Math. Univ. Carolinae 11 (1970), 559–582. MR 279827
- Freddy Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), no. 1, 53–106. MR 650816, DOI 10.1016/0022-0396(77)90136-X
- Jack Carr, Applications of centre manifold theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR 635782
- 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
- John Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), no. 1, 107–115. MR 958891, DOI 10.1090/S0002-9947-1989-0958891-1
- John Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 99–107. MR 967632, DOI 10.1017/S0143385700009366
- 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, DOI 10.1016/j.jde.2011.04.015
- 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
- MichałMisiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490–506. MR 1053617, DOI 10.1112/jlms/s2-40.3.490
- MichałMisiurewicz and Krystyna Ziemian, Rotation sets and ergodic measures for torus homeomorphisms, Fund. Math. 137 (1991), no. 1, 45–52. MR 1100607, DOI 10.4064/fm-137-1-45-52
- Jürgen K. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc. 81 (1968), 60. MR 0230498
- Floris Takens, Forced oscillations and bifurcations, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973) Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1–59. MR 0478235
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
- Email: sazanata@ime.usp.br
- 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
- MR Author ID: 111345
- Email: patrice.le-calvez@imj-prg.fr
- 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 - Communicated by: Nimish Shah
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1551-1570
- MSC (2010): Primary 37E30, 37E45
- DOI: https://doi.org/10.1090/proc/13793
- MathSciNet review: 3754341
Dedicated: This paper is dedicated to the memory of Lauro Antonio Zanata