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

Authors: Salvador Addas-Zanata and Patrice Le Calvez
Journal: Proc. Amer. Math. Soc. 146 (2018), 1551-1570
MSC (2010): Primary 37E30, 37E45
Published electronically: December 26, 2017
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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.

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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

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