Non-existence of sublinear diffusion for a class of torus homeomorphisms

In this paper, we study non-wandering homeomorphisms of the two-dimen-sional torus homotopic to the identity, whose rotation sets are non-trivial segments from (0, 0) to some totally irrational point (α, β). We show that for any r ⩾ 1, this rotation set only appears for C r diffeomorphisms satisfying some degenerate conditions. And when such a rotation set does appear, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behaviour. 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 the rotation set of some nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavours interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set.

show abstract

“…Theorem 2.10 (main result of [38]). Suppose f ∈ Homeo 0 (T 2 ), and ρ( f ) is the segment from (0, 0) to a totally irrational point (α, β).…”

Section: Bounded Deviationsmentioning

confidence: 99%

On stable and unstable behaviour of certain rotation segments

Addas-Zanata

Liu

2022

Nonlinearity

View full text Add to dashboard Cite

show abstract

“…Recently, a great deal of attention has been gathered on the problem of bounded deviation for homeomorphisms isotopic to the identity on the torus, see [2,12,17,21,28] and references therein. For homeomorphisms isotopic to the identity on the closed annulus, Conejeros and Tal showed [11] that if f is a homeomorphism on a region of instability and the rotation numbers of the boundary components lie in the interior of the rotation set, then f has uniformly bounded deviations from its rotation set.…”

Section: Introductionmentioning

confidence: 99%

Zero entropy and stable rotation sets for monotone recurrence relations

Qin

Shen

Sun

et al. 2022

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

In this paper, we show that each element in the convex hull of the rotation set of a compact invariant chain transitive set is realized by a Birkhoff solution, which is an improvement of the fundamental lemma of T. Zhou and W.-X. Qin [Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations. Math. Z.297 (2021), 1673–1692] in the study of rotation sets for monotone recurrence relations. We then investigate the properties of rotation sets assuming the system has zero topological entropy. The rotation set for a Birkhoff recurrence class is a singleton and the forward and backward rotation numbers are identical for each solution in the same Birkhoff recurrence class. We also show the continuity of rotation numbers on the set of non-wandering points. If the rotation set is upper-stable, then we show that each boundary point is a rational number, and we also obtain a result of bounded deviation.

show abstract

On non-contractible periodic orbits and bounded deviations

Liu,

Tal

2024

Nonlinearity

View full text Add to dashboard Cite

We present a dichotomy for surface homeomorphisms in the isotopy class of the identity. We show that, in the absence of a degenerate fixed point set, either there exists a uniform bound on the diameter of orbits of non-wandering points for the lifted dynamics in the universal covering space, or the map has non-contractible periodic orbits. We then use this new tool to characterize the dynamics of area preserving homeomorphisms of the torus without non-contractible periodic orbits, showing that if the fixed point set is non-degenerate, then either the lifted dynamics is uniformly bounded, or it has a single strong irrational dynamical direction.

show abstract

Non-existence of sublinear diffusion for a class of torus homeomorphisms

Abstract: We prove that, if f is a homeomorphism of the 2-torus isotopic to the identity whose rotation set is a non-degenerate segment and f has a periodic point, then it has uniformly bounded deviations in the direction perpendicular to the segment.

Cited by 3 publications

References 22 publications

On stable and unstable behaviour of certain rotation segments

On stable and unstable behaviour of certain rotation segments

Zero entropy and stable rotation sets for monotone recurrence relations

On non-contractible periodic orbits and bounded deviations

Contact Info

Product

Resources

About