Generalized Recurrence and the Nonwandering Set for Products

Abstract

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 $f\times g:X\times Y \to X\times Y$. For the generalized recurrent set $GR$, we see that $GR(f\times g)=GR(f)\times GR(g)$. For the nonwandering set $NW$, we see that $NW(f\times g)\subset NW(f)\times NW(g)$ 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.