Fixed point sets of isotopies on surfaces

Béguin, François; Crovisier, Sylvain; Roux, Frédéric Le

doi:10.4171/jems/960

Cited by 19 publications

(26 citation statements)

References 20 publications

(29 reference statements)

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“…In general, let us say that an identity isotopy of f is a maximal isotopy, if there is no fixed point of f whose trajectory is contractible relative to the fixed point set of I. A very recent result of F. Béguin, S. Crovisier and F. Le Roux (see [BCL2]) asserts that such an isotopy always exists if f is isotopic to the identity (a slightly weaker result was previously proved by O. Jaulent (see [J])).…”

Section: Isotopies Maximal Isotopiesmentioning

confidence: 99%

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Existence of non-contractible periodic orbits for homeomorphisms of the open annulus

Conejeros

Tal

2019

Math. Z.

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In this article we consider homeomorphisms of the open annulus A = R/Z × R which are isotopic to the identity and preserve a Borel probability measure of full support, focusing on the existence of non-contractible periodic orbits. Assume f is such homeomorphism such that the connected components of the set of fixed points of f are all compact. Further assume that there existsf a lift of f to the universal covering of A such that the set of fixed points off is non-empty and that this set projects into an open topological disk of A. We prove that, in this setting, one of the following two conditions must be satisfied: (1) f has non-contractible periodic points of arbitrarily large periodic, or (2) for every compact set K of A there exists a constant M (depending on the compact set) such that, ifž andf n (ž) project on K, then their projections on the first coordinate have distance less or equal to M . * The first author was supported by CNPq-Brasil. The second author was partially supported by Fapesp, CNPq-Brasil and CAPES.

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Section: Isotopies Maximal Isotopiesmentioning

confidence: 99%

“…Here we prefer to follow [BCL2], because Jaulent's Theorem about existence of maximal isotopies cannot be stated in the following natural form.…”

Section: Isotopies Maximal Isotopiesmentioning

confidence: 99%

Existence of non-contractible periodic orbits for homeomorphisms of the open annulus

Conejeros

Tal

2019

Math. Z.

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“…In this section, we will recall some results about the isotopies of surface homeomorphisms due to Jaulent [6] and Béguin, Crovisier and Le Roux [1].…”

Section: Unlinked Sets and Maximal Isotopiesmentioning

confidence: 99%

“…More precisely, an identity isotopy I of a surface homeomorphism f is a continuous family of homeomorphisms (f t ) t∈[0,1] with f 0 = Id and f 1 = f , and a maximal isotopy is an identity isotopy I = (f t ) t∈[0, 1] of f such that f does not have any fixed point whose trajectory along I is contractible in M \ Fix(I), where Fix(I) = ∩ t∈[0,1] Fix(f t ) is the fixed points set of I. To such an isotopy, one can associate transverse foliations [10], i.e.…”

Section: Introduction and Definitionsmentioning

confidence: 99%

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Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms

Yan

2018

Discrete &Amp; Continuous Dynamical Systems - A

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The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal isotopy such that we can give a fine description of the dynamics of its transverse foliation. We will define a sort of identity isotopies: torsion-low isotopies. In particular, when f is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy I of f is torsion-low if and only if for every point z fixed along the isotopy, the (real) rotation number ρ(I, z) (which is well defined when one blows up f at z) is contained in (−1, 1). We will prove the existence of torsion-low maximal isotopies, and will deduce the local dynamics of the transverse foliations of any torsion-low maximal isotopy near any isolated singularity.

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Some results of Hamiltonian homeomorphisms on closed aspherical surfaces

Wang

2020

Advances in Mathematics

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Fixed point sets of isotopies on surfaces

Cited by 19 publications

References 20 publications

Existence of non-contractible periodic orbits for homeomorphisms of the open annulus

Existence of non-contractible periodic orbits for homeomorphisms of the open annulus

Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms

Some results of Hamiltonian homeomorphisms on closed aspherical surfaces

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