  We study Nash equilibria for the deterministic ergodic N-players game. We
introduce pure strategies, mixed strategies and Nash equilibria associated with
those. We show that a Nash equilibrium in mixed strategies exists and it is a
Mather measure for the Lagrangian system defined by the cost functional. In
conclusion, we show that the mean field limit of the N-players game is
described by the ergodic PDE's system for a continuum of players.
