Rational polygons as rotation sets of generic homeomorphisms of the two torus

Passeggi, Alejandro

doi:10.1112/jlms/jdt040

Cited by 24 publications

(36 citation statements)

References 16 publications

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“…The proofs also work without the area-preservation, and are considerably simpler than those from [Pas14]. We note that the latter article relies heavily on the genericity of Axiom A dynamics in Homeo 0 (T 2 ), a fact which no longer holds in the area-preserving setting.…”

Section: Introductionmentioning

confidence: 99%

“…The set of all homeomorphisms with a stable rotation set is open and dense in H. Moreover, the rotation set of every such homeomorphism is a convex polygon with rational vertices, and in the area-preserving case this polygon has nonempty interior. This is a bit surprising: one could have expected that in the area-preserving case, there is generically a wide set of examples of rotation sets, and that they are unstable, as the phenomena used in the proofs of [Pas14] do not hold, and as the dynamical behaviour of generic conservative homeomorphisms is much richer than that of arbitrary generic homeomorphisms (compare [AHK03] with [Gui12]). …”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

Stability of the rotation set of area-preserving toral homeomorphisms

Guihéneuf

Koropecki

2017

Nonlinearity

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“…Here the rotation vectors are the averages of f with respect to the possible T -invariant probability measures on X. In the paper [19] Passeggi proves, in some sense, the result reverse to the content of [11]: It is shown there that the rotation set of a topologically generic homeomorphism of T 2 is a rational polygon.…”

Section: Introductionmentioning

confidence: 92%

Homotopical complexity of a billiard flow on the 3D flat torus with two cylindrical obstacles

Moxley¹,

Simányi²

2017

Ergod. Th. Dynam. Sys.

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We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two disjoint and orthogonal toroidal (cylindric) scatterers removed from it.The natural habitat for these objects is the infinite cone erected upon the Cantor set Ends(G) of all "ends" of the hyperbolic group G = π 1 (Q). An element of Ends(G) describes the direction in (the Cayley graph of) the group G in which the considered trajectory escapes to infinity, whereas the height function s (s ≥ 0) of the cone gives us the average speed at which this escape takes place.The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding √ 3, and in any direction e ∈ Ends(π 1 (Q)) the escape is feasible with any prescribed. This means that the radial upper and lower bounds for the rotation set R are actually pretty close to each other. Furthermore, we prove the convexity of the set AR of constructible rotation vectors, and that the set of rotation vectors of periodic orbits is dense in AR. We also provide effective lower and upper bounds for the topological entropy of the studied billiard flow.

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“…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. Years later, Passeggi [28] proved that maps whose rotation sets are rational polygons form an open and dense set among all torus homeomorphisms homotopic to the identity. Moreover, the rotation set of any axiom A diffeomorphism on a torus is a rational polygon.…”

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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Rational polygons as rotation sets of generic homeomorphisms of the two torus

Cited by 24 publications

References 16 publications

Stability of the rotation set of area-preserving toral homeomorphisms

Stability of the rotation set of area-preserving toral homeomorphisms

Homotopical complexity of a billiard flow on the 3D flat torus with two cylindrical obstacles

Entropy and rotation sets: A toy model approach

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