A finite dimensional approach to Bramham’s approximation theorem

Calvez, Patrice Le

doi:10.5802/aif.3061

Cited by 7 publications

(3 citation statements)

References 4 publications

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“…That is, there exists a sequence of iterates f n j that converges to the identity in the C 0 -topology as n j Ñ 8. Le Calvez [25] proved similar results for C 1 irrational pseudo-rotation f whose restriction to S 1 is C 1 conjugate to a rotation.…”

Section: Introductionmentioning

confidence: 58%

“…Bramham already showed in [8] that any C 8 irrational pseudo-rotation f of the disk with super Liouville rotation number is C 0 rigid, meaning that f is the C 0 -limit of a sequence of periodic diffeomorphisms. More recently Le Calvez [25] proved that any C 1 irrational pseudo-rotation which is C 1 conjugated to a rotation on the boundary is C 0 rigid. These results go as follows.…”

Section: Examplesmentioning

confidence: 99%

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The Calabi invariant for Hamiltonian diffeomorphisms of the unit disk

Joly

2021

Preprint

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In this article, we study the Calabi invariant on the unit disk usually defined on compactly supported Hamiltonian diffeomorphisms of the open disk. In particular we extend the Calabi invariant to the group of C 1 diffeomorphisms of the closed disk which preserves the standard symplectic form. We also compute the Calabi invariant for some diffeomorphisms of the disk which satisfies some rigidity hypothesis.

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Section: Introductionmentioning

confidence: 58%

Section: Examplesmentioning

confidence: 99%

The Calabi invariant for Hamiltonian diffeomorphisms of the unit disk

Joly

2021

Preprint

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“…In a sense, as far as surfaces are concerned, the two theories appear to be equivalent: much of what can be done via one theory can also be achieved via the other. As examples of this phenomenon, one could point to proofs of the Arnol'd conjecture and recent articles by Bramham [2,3] and Le Calvez [27]. A prominent missing link from this hypothetical equivalence is the spectral invariant c which to this date has had no analogue in Le Calvez's theory.…”

Section: Introductionmentioning

confidence: 99%

Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

2016

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Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces [25,26], we introduce a dynamical invariant, denoted by N , for Hamiltonians of any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces.Along the way, we obtain several results of independent interest: We show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with N on autonomous Hamiltonians thus establishing a certain uniqueness result for spectral invariants, we obtain a "Max Formula" for spectral invariants on aspherical manifolds, give a very simple description of the Entov-Polterovich quasi-state on aspherical surfaces and characterize the heavy and super-heavy subsets of such surfaces.Formal Spectral Invariants: Although the following definition makes sense on any symplectic manifold we will restrict our attention here to the case of a surface Σ which is either the plane R 2 or is closed and aspherical. Definition 3. A function c : C ∞ ([0, 1] × Σ) → R is a formal spectral invariant if it satisfies the following four axioms: 1. (Spectrality) c(H) ∈ spec(H) for all H ∈ C ∞ ([0, 1]×Σ), where spec(H), the spectrum of H, is the set of critical values of the Hamiltonian action, that is, the set of actions of fixed points of φ 1 H .

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On mixing diffeomorphisms of the disc

et al. 2019

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We prove that a real analytic pseudo-rotation f of the disc or the sphere is never topologically mixing. When the rotation number of f is of Brjuno type, the latter follows from a KAM theorem of Rüssmann on the stability of real analytic elliptic fixed points. In the non-Brjuno case, we prove that a pseudo-rotation of class C k , k ≥ 2, is C k−1 -rigid using the simple observation, derived from Franks' Lemma on free discs, that a pseudo-rotation with small rotation number compared to its C 1 (or Hölder) norm must be close to Identity.From our result and a structure theorem by Franks and Handel (on zero entropy surface diffeomorphisms) it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.In our proof we need an a priori limit on the growth of the derivatives of the iterates of a pseudo-rotation that we obtain via an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f , that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential.

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A finite dimensional approach to Bramham’s approximation theorem

Cited by 7 publications

References 4 publications

The Calabi invariant for Hamiltonian diffeomorphisms of the unit disk

The Calabi invariant for Hamiltonian diffeomorphisms of the unit disk

Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

On mixing diffeomorphisms of the disc

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