On a conjecture of Anosov

Wang, Wei

doi:10.1016/j.aim.2012.04.006

Cited by 35 publications

(43 citation statements)

References 27 publications

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“…In this setting, the right analog of the dynamical convexity condition is the requirement that µ − ≥ n − 1 for all closed Reeb orbits of α. For the unit sphere bundle of a Riemannian or Finsler metric, this requirement is satisfied, for instance, when the metric meets certain curvature pinching conditions; see, e.g., [AM14,DLW,HP,Ra04,Wa12]. Along the lines of Theorem 1.5, we have the following result.…”

Section: Introduction and Main Resultsmentioning

confidence: 90%

Lusternik–Schnirelmann theory and closed Reeb orbits

Ginzburg

Gürel

2019

Math. Z.

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We develop a variant of Lusternik-Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. As an application, we prove new multiplicity results for simple closed Reeb orbits on the standard contact sphere, the unit cotangent bundle to the sphere and some other contact manifolds. We also show that the lower Conley-Zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of Long and Zhu. This is the second, combinatorial ingredient in the proof of the multiplicity results.

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Section: Introduction and Main Resultsmentioning

confidence: 90%

Lusternik–Schnirelmann theory and closed Reeb orbits

Ginzburg

Gürel

2019

Math. Z.

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“…The same author proved in [36] that for every Finsler n-dimensional sphere S n for n ≥ with reversibility λ and flag curvature K satisfying (λ/( + λ)) < K ≤ , either there exist infinitely many prime closed geodesics or there exist [n/ ] − closed geodesics possessing irrational mean indices. Furthermore, assuming that the metric F is bumpy, he showed in [35] that there exist [(n + )/ ] closed geodesics on (S n , F). Also, in [35], he showed that for every bumpy Finsler metric F on S n satisfying ( / )(λ/( + λ)) < K ≤ , there exist two prime elliptic closed geodesics provided the number of closed geodesics on (S n , F) is finite.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Furthermore, assuming that the metric F is bumpy, he showed in [35] that there exist [(n + )/ ] closed geodesics on (S n , F). Also, in [35], he showed that for every bumpy Finsler metric F on S n satisfying ( / )(λ/( + λ)) < K ≤ , there exist two prime elliptic closed geodesics provided the number of closed geodesics on (S n , F) is finite.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Closed Geodesics on Positively Curved Finsler 3-Spheres

Duan

Liu

2016

Advanced Nonlinear Studies

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Abstract:In [33], Wang proved that for every Finsler three-dimensional sphere (S , F) with reversibility λ and flag curvature K satisfying (λ/( + λ)) < K ≤ , there exist at least three distinct closed geodesics. In this paper, we prove that for every Finsler three-dimensional sphere (S , F) with reversibility λ and flag curvature K satisfying ( / )(λ/( + λ)) < K ≤ with λ < , if there exist exactly three prime closed geodesics, then two of them are irrationally elliptic and the third one is infinitely degenerate.

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“…There are many works using Morse theory and index theory to study the closed geodesics and Anosov Conjecture in Finsler geometry, assuming a pinch condition for the flag curvature, non-degenerating property for all closed geodesics, or using the speciality of low dimensions. See for example [7] [11] [12] [13] [15] [18] [19]. From the geometrical point of view, it was much later that people noticed that Katok metrics are Randers metrics on spheres with constant flag curvature [16].…”

Section: Introductionmentioning

confidence: 99%

Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics

2019

Pacific J. Math.

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In this paper, we consider a Finsler sphere (M, F ) = (S n , F ) with the dimension n > 1 and the flag curvature K ≡ 1. The action of the connected isometry group G = I o (M, F ) on M , together with the action of T = S 1 shifting the parameter t ∈ R/Z of the closed curve c(t), define an action ofĜ = G × T on the free loop space λM of M . In particular, for each closed geodesic, we have aĜ-orbit of closed geodesics. We assume the Finsler sphere (M, F ) described above has only finite orbits of prime closed geodesics. Our main theorem claims, if the subgroup H of all isometries preserving each close geodesic has a dimension m, then there exists m geometrically distinct orbits B i of prime closed geodesics, such that for each i, the union B i of geodesics in B i is a totally geodesic sub-manifold in (M, F ) with a non-trivial H o -action. This theorem generalizes and slightly refines the one in a previous work, which only discussed the case of finite prime closed geodesics. At the end, we show that, assuming certain generic conditions, the Katok metrics, i.e. the Randers metrics on spheres with K ≡ 1, provide examples with the sharp estimate for our main theorem.Mathematics Subject Classification (2000): 22E46, 53C22, 53C60.

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On a conjecture of Anosov

Abstract: In this paper, we prove that for every bumpy Finsler n-sphere (S n , F ) with reversibility λ and flag curvature K satisfying λ λ+1 2 < K ≤ 1, there exist 2[ n+1 2 ] prime closed geodesics. This gives a confirmed answer to a conjecture of D. V. Anosov [Ano] in 1974 for a generic case.

Cited by 35 publications

References 27 publications

Lusternik–Schnirelmann theory and closed Reeb orbits

Lusternik–Schnirelmann theory and closed Reeb orbits

Closed Geodesics on Positively Curved Finsler 3-Spheres

Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics

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