Diffeomorphisms in symplectic category posses certain rigidity properties

(symplectic camel, Gromov non-squeezing theorem). An important

manifestation of rigidity is given by the conjectures posed by V. Arnold

describing a lower bound for the number of fixed points of a Hamiltonian

diffeomorphism h (i.e. symplectic diffeomorphism Hamiltonian isotopic to

identity) of a compact symplectic manifold. These lower bounds

are greater than what topological arguments could predict.

Arnold Conjectures present a difficult problem and motivated a lot of important

research in symplectic geometry. The case of C¹-small Hamiltonian diffeomorphisms

is easy to prove. Generalization of these ideas gave rise to the technique of generating

functions.

In this talk I will introduce generating functions and sketch a proof of

Arnold Conjecture for a complex projective space.

Then I will formulate a version of Arnold Conjecture in contact geometry

setting describing the minimal number of translated points of a contact

isotopy (posted by S. Sandon) and describe a current work in progress, with

G. Granja, Y. Karshon and S. Sandon, aimed to prove this conjecture for

lens spaces and other contact toric manifolds.