ON ITERATED TRANSLATED POINTS FOR CONTACTOMORPHISMS OF ℝ²ⁿ⁺¹ AND ℝ²ⁿ × S¹

Abstract

Abstract. A point q in a contact manifold is called a translated point for a contactomorphism φ with respect to some fixed contact form if φ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points has an interpretation in terms of Reeb chords between Legendrian submanifolds, and can be seen as a special case of the problem of leafwise coisotropic intersections. For a compactly supported contactomorphism φ of R 2n+1 or R 2n × S 1 contact isotopic to the identity, existence of translated points follows immediately from Chekanov's theorem on critical points of quasi-functions and Bhupal's graph construction. In this article we prove that if φ is positive then there are infinitely many non-trivial geometrically distinct iterated translated points, i.e. translated points of some iteration φ k . This result can be seen as a (partial) contact analogue of the result of Viterbo on existence of infinitely many iterated fixed points for compactly supported Hamiltonian symplectomorphisms of R 2n , and is obtained with generating functions techniques in the setting of [S11].

Leafwise Fixed Points for $C^0$-Small Hamiltonian Flows

“…So far solutions to Problem 1 have been found under restrictive assumptions on ϕ or N, see [Mo,Ban,EH,Ho2,Gi,Dr,Gü,Zi1,Zi2,AF1,AF2,AF3,AMo,AMc,Bae,Ka1,Ka2,MMP,Sa1,Sa2]. In [Mo, Ban] J. Moser and A. Banyaga assumed that the restriction ϕ| N is C 1 -close to the inclusion N → M. In most other results this condition was relaxed to the condition that ϕ be Hofer close to the identity.…”

Givental’s Non-linear Maslov Index on Lens Spaces

“…In these works the role played by discriminant and translated points is made more explicit, and appears to be similar to the one described in [Gi90]. Recall from [Sa11a,Sa12] that a point p of a contact manifold (V, ξ) is said to be a translated point of a contactomorphism φ with respect to a contact form α for ξ if p is a discriminant point of ϕ α −η • φ for some real number η (called the time-shift ), where ϕ α t denotes the Reeb flow. In terms of the Legendrian graph gr(φ) in the contact product V × V × R, discriminant and translated points correspond, respectively, to intersections and Reeb chords between gr(φ) and the diagonal ∆ × {0} = gr(id).…”

Maximum principles in symplectic homology

“…We call the minimal (in absolute value) such T the period of x (the word "minimal" is included to deal with the case where the translated point lies on a closed Reeb orbit of α.) This notion was introduced by Sandon in [22], although it can be seen as a special case of a leaf-wise intersection point as introduced by Moser in [13].…”

“…Given a contactomorphism ϕ : Σ → Σ of a contact manifold (Σ, α), a translated point of ϕ is a point x such that ϕ(x) and x both lie on the same Reeb orbit of α, and such that ϕ * α| x = α x . This notion was introduced by Sandon in [22], although it can be seen as a special case of a leaf-wise intersection point as introduced by Moser in [13]. Using Rabinowitz Floer homology, the first author and Albers proved:…”