Forcing theory for transverse trajectories of surface homeomorphisms

Calvez, Patrice Le; Tal, Fabio Armando

doi:10.1007/s00222-017-0773-x

Cited by 41 publications

(120 citation statements)

References 27 publications

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“…In this article, improving [JT16], we show that a possible counter example must exhibit unbounded deviations in the complementary direction of the supporting line of the interval ρ(F ). This turns to be quite suggesting as it is shown in several cases that having two different rotation vectors, is an obstruction for deviations [Dáv16,CT15].…”

Section: Introductionmentioning

confidence: 68%

Deviations in the Franks–Misiurewicz conjecture

Passeggi¹,

Sambarino²

2019

Ergod. Th. Dynam. Sys.

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Section: Introductionmentioning

confidence: 68%

Deviations in the Franks–Misiurewicz conjecture

Passeggi¹,

Sambarino²

2019

Ergod. Th. Dynam. Sys.

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“…As a consequence, it works for surface homeomorphisms without any differentiability assumption. This might be of interest in relation with some recent works related to the Franks-Misiurewicz conjecture (see [4,3], and the example of Avila quoted in these papers). * S.C was partially supported by the ERC project 692925 NUHGD.…”

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confidence: 80%

Projective transformations of rotation sets

2019

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We give a new proof and extend a result of J. Kwapisz: whenever a set C is realized as the rotation set of some torus homeomorphism, the image of C under certain projective transformations is also realized as a rotations set.The concept of rotation set, introduced by M. Misiurewicz and K. Ziemian in [5], is one of the most important tools to study the global dynamics of homeomorphisms of the torus T 2 . If f is a homeomorphism of T 2 isotopic to the identity, and F is a lift of f to R 2 , the rotation set of F is a compact convex subset of the plane which describes "at what speeds and in what directions the orbits of f rotate around the torus". One of the main problems in the theory is to determine which compact convex subsets of R 2 can be realized as the rotations sets of some torus homeomorphisms. For compact convex subsets with empty interiors (i.e. singletons and segments), a conjectural answer to the problem has been formulated by J. Franks and M. Misiurewicz (see [1]). Fifteen years ago, J. Kwapicz has introduced a technical tool which allows to simplify the problem. Namely, he observed that, if a compact convex set C ⊂ R 2 is realized as the rotation of a certain torus diffeomorphism, and if a projective transformation L maps C to a bounded set of the plane, then L(C) can be realized as the rotation of another torus diffeomorphism (see [2, section 2]).Kwapisz's proof requires to consider the suspension of the initial torus homeomorphism, and to apply a theorem of D. Fried to find a new surface of section for this flow, in the appropriate cohomology class. Fried's theorem works only for C 1 flows; this forces Kwapisz to consider only rotation sets of C 1 diffeomorphisms, whereas the natural setting for his result would be rotation sets of homeomorphisms. The purpose of the present note is to provide a more elementary proof of Kwapisz's result. Our proof remains at the level of surfaces homeomorphisms, i.e. does not require to consider a flow on a three-dimensional manifold. It does not make use of Fried's theorem (in some sense, we replace it by the more classical fact that the only surface with fundamental group isomorphic to Z 2 is the torus T 2 ). As a consequence, it works for surface homeomorphisms without any differentiability assumption. This might be of interest in relation with some recent works related to the Franks-Misiurewicz conjecture (see [4,3], and the example of Avila quoted in these papers). * S.C was partially supported by the ERC project 692925 NUHGD.

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“…There are many interesting problems and results on the relation between the rotation set and the dynamics (see, e.g., [MZ89,FM90,LM91,LeCT15,Koc16,KPS16]). It is shown in [LM91,Fra89] that a torus homeomorphism which is isotropic the identity, and has a rotation set with nonempty interior must have positive topological entropy.…”

Section: Introductionmentioning

confidence: 99%

The rigidity of pseudo-rotations on the two-torus and a question of Norton-Sullivan

Wang

Zhang

2018

Geom. Funct. Anal.

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We show that under certain boundedness condition, a C r conservative irrational pseudo-rotations on T 2 with a generic rotation vector is C r−1 -rigid. We also obtain C 0 -rigidity for Hölder pseudo-rotations with similar properties. These provide a partial generalisation of the main results in [Bra15, AFLXZ15].We then use these results to study conservative irrational pseudo-rotations on T 2 with a generic rotation vector that is semi-conjugate to a translation via a semi-conjugacy homotopic to the identity. We show that the conservative centralizers of any such diffeomorphism is isomorphic to a uncountable subgroup of R 2 /Z 2 . In connection with a question of Alec Norton and Dennis Sullivan, we describe the topologically linearizable maps within this class using the topology of the conservative centralizer group. In the minimal case, we obtain a precise characterization of topological linearizability for all totally irrational vectors.We also construct a C ∞ conservative and minimal totally irrational pseudo-rotation diffeomorphism that is semi-conjugate to a translation, but is topologically nonlinearizable. This gives a negative answer to the question of Norton and Sullivan in the C ∞ category.

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Forcing theory for transverse trajectories of surface homeomorphisms

Cited by 41 publications

References 27 publications

Deviations in the Franks–Misiurewicz conjecture

Deviations in the Franks–Misiurewicz conjecture

Projective transformations of rotation sets

The rigidity of pseudo-rotations on the two-torus and a question of Norton-Sullivan

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