  The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of
${\mathbb C}{\mathbb P}^n$ with the minimal possible number of periodic points
(equal to $n+1$ by Arnold's conjecture), called here Hamiltonian
pseudo-rotations. We prove several results on the dynamics of pseudo-rotations
going beyond periodic orbits, using Floer theoretical methods. One of these
results is the existence of invariant sets in arbitrarily small punctured
neighborhoods of the fixed points, partially extending a theorem of Le Calvez
and Yoccoz and Franks to higher dimensions. The other is a strong variant of
the Lagrangian Poincar\'e recurrence conjecture for pseudo-rotations. We also
prove the $C^0$-rigidity of pseudo-rotations with exponentially Liouville mean
index vector. This is a higher-dimensional counterpart of a theorem of Bramham
establishing such rigidity for pseudo-rotations of the disk.
