  The aim of the paper is three-fold. We begin by proving a formula, both
global and local versions, relating the number of periodic orbits of an
iterated map and the Lefschetz numbers, or indices in the local case, of its
iterations. This formula is then used to express the mean Euler characteristic
(MEC) of a contact manifold in terms of local, purely topological, invariants
of closed Reeb orbits, without any non-degeneracy assumption on the orbits.
Finally, turning to applications of the local MEC formula to dynamics, we use
it to reprove a theorem asserting the existence of at least two closed Reeb
orbits on the standard contact three-sphere (by Cristofaro-Gardiner and
Hutchings in the most general form) and the existence of at least two closed
geodesics for a Finsler metric on the two-sphere (Bangert and Long).
