Generalized Recurrence and the Nonwandering Set for Products

For continuous maps of compact metric spaces $f:X\to X$ and $g:Y\to Y$ and for various notions of topological recurrence, we study the relationship between recurrence for $f$ and $g$ and recurrence for the product map $f\times g:X\times Y \to X\times Y$\mathrm{.}Forthegeneralizedrecurrentset$GR$,weseethat$GR(f\times g)=GR(f)\times GR(g)$\mathrm{.}Forthenonwanderingset$NW$,weseethat$NW(f\times g)\subset NW(f)\times NW(g)$ 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 Ma~n'e set.