Report
On Stable and Unstable Behaviours of Certain Rotation Segments
| العنوان: | On Stable and Unstable Behaviours of Certain Rotation Segments |
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| المؤلفون: | Addas-Zanata, Salvador, Liu, Xiao-Chuan |
| سنة النشر: | 2019 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Dynamical Systems |
| الوصف: | In this paper, we study non-wandering homeomorphisms of the two torus in the identity homotopy class, whose rotation sets are non-trivial line segments from $(0,0)$ to some totally irrational vector $(\alpha,\beta)$. We show this rotation set is in fact a non-generic phenomenon for any $C^r$ diffeomorphisms, with $r \geq 1$. When such a rotation set does happen, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behavior. More precisely, the dynamics admits bounded deviation along the direction $-(\alpha,\beta)$ in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small $C^0$-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism $f$ with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by rotation sets of nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavors interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set. Comment: 8 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1903.08703 |
| رقم الانضمام: | edsarx.1903.08703 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |