  Replicator dynamics, the continuous-time analogue of Multiplicative Weights
Updates, is the main dynamic in evolutionary game theory. In simple
evolutionary zero-sum games, such as Rock-Paper-Scissors, replicator dynamic is
periodic \cite{zeeman1980population}, however, its behavior in higher
dimensions is not well understood. We provide a complete characterization of
its behavior in zero-sum evolutionary games. We prove that, if and only if, the
system has an interior Nash equilibrium, the dynamics exhibit Poincar\'{e}
recurrence, i.e., almost all orbits come arbitrary close to their initial
conditions infinitely often. If no interior equilibria exist, then all interior
initial conditions converge to the boundary. Specifically, the strategies that
are not in the support of any equilibrium vanish in the limit of all orbits.
All recurrence results furthermore extend to a class of games that generalize
both graphical polymatrix games as well as evolutionary games, establishing a
unifying link between evolutionary and algorithmic game theory. We show that
two degrees of freedom, as in Rock-Paper-Scissors, is sufficient to prove
periodicity.
