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arXiv · 1903.08703

On Stable and Unstable Behaviours of Certain Rotation Segments

Salvador Addas-Zanata·Xiao-Chuan Liu
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Abstract

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 $(α,β)$. 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 $-(α,β)$ 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.

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Salvador Addas-Zanata, Xiao-Chuan Liu. 2021-12-24. On Stable and Unstable Behaviours of Certain Rotation Segments. https://doi.org/10.1088/1361-6544%2Fac8f0d

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