Topological dynamic classification of duopoly games

We classify Cournot maps on the unit square by using topological dynamics. More precisely, we prove that the following properties are equivalent: (1) zero topological entropy, (2) UR(F)=R(F), (3) type less than or equal to 2∞ and (4) AP(F)={(x,y) I2:limn→∞F2n(x,y)=(x,y)}. These results allow us to decide when the behavior of these two-dimensional dynamical systems is complicated.