Generalized Recurrence and the Nonwandering Set for Products

TL;DR

This paper investigates how various notions of recurrence, including generalized recurrence and nonwandering sets, behave under product maps in dynamical systems, establishing key equalities and conditions for these properties.

Contribution

It proves that the generalized recurrent set of a product equals the product of the sets, and characterizes when the nonwandering set of a product equals the product of the sets.

Findings

GR(f×g) = GR(f) × GR(g)

NW(f×g) ⊆ NW(f) × NW(g) with conditions for equality

Analysis of recurrence for chain, strong chain, and Mañé sets

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.