Sur les groupes compacts de transformations topologiques des surfaces

Kerékjártó, B. v.

doi:10.1007/bf02392252

Cited by 12 publications

(14 citation statements)

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“…Notice that the group in discussion is torsion free, due to a classical result of Kerékjártó [76] (see also [77]). It is worth to stress that it is even unknown whether this group satisfies the UPP.…”

Section: Cones and Orders On Groupsmentioning

confidence: 99%

Group Actions on 1-Manifolds: A List of Very Concrete Open Questions

Navas

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Over the last four decades, group actions on manifolds have deserved much attention by people coming from different fields, as for instance group theory, low-dimensional topology, foliation theory, functional analysis, and dynamical systems. This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear panorama on the subject arises from the lecture.

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Section: Cones and Orders On Groupsmentioning

confidence: 99%

Group Actions on 1-Manifolds: A List of Very Concrete Open Questions

Navas

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…Proof of theorem 1. It is a classical result [7,9,28] that if f ∈ Homeo + (D) satisfies f n = id D for some n ∈ N, then there exists g ∈ Homeo + (D) and q ∈ {0, 1, . .…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Hamiltonians on the disk H t − := H − (t, ·) : D → R has time-one map ϕ, and Rot(ϕ, H − ) = α ∈ R. Define H + by (28) H + (τ, z) := πα|z| 2 + C for some constant C ≤ 0 chosen so that maxH + < min H − . Then the two pairs of differential forms H ± = (ω ± , λ ± ), on Z, given by…”

Section: Construction Of the Finite Energy Foliationsmentioning

confidence: 99%

“…Each v c,z has image bounded from above, so we have to insert one of them into the negative end of our almost complex manifold (W = R × Z,Ĵ ). So the main difference is that from the start we reverse the roles of H − and H + , this time picking a positive constant C in (28) so that min H − > maxH + still holds. Then the remaining steps are exactly analogous, and in the final foliation the curves with boundary have boundary index ⌈α⌉ instead, as the curves in (31) do.…”

Section: 4mentioning

confidence: 99%

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Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

Bramham

2015

Ann. Math.

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We prove that every C ∞ -smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic C ∞ -smooth diffeomorphisms. In particular every smooth irrational pseudo-rotation can be C 0 -approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.Date: April 23, 2012. 1 1.2. Idea of the proof. Pick a closed loop of Hamiltonians H t : D → D, over t ∈ R/Z, which generate a symplectic isotopy whose time-one map is ϕ. Denote the 1-periodic path of Hamiltonian vector fields on the disk by X Ht . For each n ∈ N equip Z n = R/nZ × D with coordinates (τ, z). Then the vector field R n (τ, z) = ∂ τ + X Ht (z) defines a flow on Z n with time-one map ϕ and first return map ϕ n .Consider the 4-manifold W n := R × Z n with the unique almost complex structure satisfyingwhere ∂ R is the vector field dual to the R-coordinate on W n . Then (W n , J n ) is a so called cylindrical, symmetric, almost complex manifold. That is, it is compatible in a precise way with the necessary symplectic structures for the compactness framework from symplectic field theory [4] to apply to J n -holomorphic curves.

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“…Since the seminal work of L. E. Brouwer nearly 100 years ago, they draw attentions of many mathematicians ( [3], [8], [4], [5] [6], [11]). Nowadays there still remains interesting problems about them.…”

Section: Introductionmentioning

confidence: 99%