Every convex polygon with rational vertices is a rotation set

Kwapisz, Jarosław

doi:10.1017/s0143385700006787

Cited by 40 publications

(44 citation statements)

References 4 publications

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“…About (2), it is known that the rotation set is compact and convex [MZ89], every convex polygon of rational vertices is the rotation set of some homeomorphism [Kwa92], but there are non-polygonal examples [Kwa95,BdCH16] (although the known ones are "almost polygonal": they have countably many extremal points).…”

Section: Introductionmentioning

confidence: 99%

Stability of the rotation set of area-preserving toral homeomorphisms

Guihéneuf

Koropecki

2017

Nonlinearity

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Abstract. We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are C 0 -generic and have bounded rotational deviations (even for pseudo-orbits). The results hold both in the area-preserving setting and in the general setting. When the rotation set is stable, we give explicit estimates on the type of rationals that may appear as vertices of rotation sets in terms of the stability constants.

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Section: Introductionmentioning

confidence: 99%

Stability of the rotation set of area-preserving toral homeomorphisms

Guihéneuf

Koropecki

2017

Nonlinearity

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“…As we mentioned, the only known examples are polygons (any rational polygon is realized as a rotation set [25]) or 'infinite polygons' with a countable set of extremal points. As we mentioned, the only known examples are polygons (any rational polygon is realized as a rotation set [25]) or 'infinite polygons' with a countable set of extremal points.…”

Section: Introductionmentioning

confidence: 99%

On Annular Maps of the Torus and Sublinear Diffusion

Dávalos¹

2016

J. Inst. Math. Jussieu

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A classical article by Misiurewicz and Ziemian (J. Lond. Math. Soc. 40(2) (1989), 490-506) classifies the elements in Homeo 0 (T 2 ) by their rotation set ρ, according to wether ρ is a point, a segment or a set with nonempty interior. A recent classification of nonwandering elements in Homeo 0 (T 2 ) by Koropecki and Tal was given in (Invent. Math. 196 (2014), 339-381), according to the intrinsic underlying ambient space where the dynamics takes place: planar, annular and strictly toral maps. We study the link between these two classifications, showing that, even abroad the nonwandering setting, annular maps are characterized by rotation sets which are rational segments. Also, we obtain information on the sublinear diffusion of orbits in the-not very well understood-case that ρ has nonempty interior.

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“…In [22] Kwapisz considers the opposite question: which subsets of R 2 can arise as rotation sets for torus homeomorphisms. He proved that any polygon whose vertices are at rational points in the plane can be obtained as the rotation set of some homeomorphism on the two-torus.…”

Section: Rotation Theorymentioning

confidence: 99%

Entropy and rotation sets: A toy model approach

Kucherenko

Wolf

2016

Commun. Contemp. Math.

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Given a continuous dynamical system f on a compact metric space X and a continuous potential Φ : X → R m , the generalized rotation set is the subset of R m consisting of all integrals of Φ with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measure-theoretic entropies over all invariant measures whose integrals produce that point. In this paper, we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at a particular exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy.

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Every convex polygon with rational vertices is a rotation set

Abstract: We prove that every convex polygon in ℝ2with vertices in ℚ2is a rotation set for some isotopic to identity homeomorphism of the two dimensional terms.

Cited by 40 publications

References 4 publications

Stability of the rotation set of area-preserving toral homeomorphisms

Stability of the rotation set of area-preserving toral homeomorphisms

On Annular Maps of the Torus and Sublinear Diffusion

Entropy and rotation sets: A toy model approach

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