Approximate Identities and Lagrangian Poincar\'e Recurrence

In this note we discuss three interconnected problems about dynamics of Hamiltonian or, more generally, just smooth diffeomorphisms. The first two concern the existence and properties of the maps whose iterations approximate the identity map with respect to some norm, e.g., $C^1$- or $C^0$-norm for general diffeomorphisms and the $\gamma$-norm in the Hamiltonian case, and the third problem is the Lagrangian Poincar\'e recurrence conjecture.