Chern-Weil theory and the group of strict contactomorphisms

In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group U(n+1) is homotopically essential in the group of contactomorphisms of the standard contact sphere S^(2n+1) and prove that in the case of the 3-sphere the contactomorphism group is in fact homotopy equivalent to U(2). In the second part of the paper we focus on the group of strict contactomorphisms -- using the framework of Chern-Weil theory we introduce and study contact characteristic classes analogous to the Reznikov Hamiltonian classes in symplectic topology. We carry out several explicit calculations illustrating non-triviality of the contact classes.