Generalized recurrence and the nonwandering set for products

For continuous maps of compact metric spaces f:X→X and g:Y→Y and for various notions of topological recurrence, we study the relationship between recurrence for f and g and recurrence for the product map f×g:X×Y→X×Y. For the generalized recurrent set GR, we see that GR(f×g)=GR(f)×GR(g). For the nonwandering set NW, we see that NW(f×g)⊂NW(f)×NW(g) and give necessary and sufficient conditions on f for equality for every g. We also consider product recurrence for the chain recurrent set, the strong chain recurrent set, and the Mañé set.