Abstract. A point q in a contact manifold (M, ξ) 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. In this article we discuss a version of the Arnold conjecture for translated points of contactomorphisms and, using generating functions techniques, we prove it in the case of spheres (under a genericity assumption) and projective spaces.

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on R 2n × S 1 and RP 2n+1 . On the other hand we also show by elementary arguments that the discriminant metric is bounded for the standard contact structures on R 2n+1 and S 2n+1 . As an application of these results we get that the contact fragmentation norm is unbounded for R 2n × S 1 and RP 2n+1 . By elaborating on the construction of the discriminant metric we then define an integer-valued bi-invariant pseudo-metric, that we call the oscillation pseudo-metric, which is non-degenerate if and only if the contact manifold is orderable in the sense of Eliashberg and Polterovich and, in this case, it is compatible with the partial order. Finally we define the discriminant and oscillation lengths of a Legendrian isotopy, and prove that they are unbounded for T * B × S 1 for any closed manifold B, for RP 2n+1 and for some 3-dimensional circle bundles.1 Recall that any conjugation-invariant norm ν : G → [0, +∞) on a group G induces a bi-invariant metric dν on G by defining dν (f, g) = ν(f −1 g). 2 If M is not compact, we will consider the discriminant norm on the universal cover of the identity component of the group of compactly supported contactomorphisms.

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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].