Nonwandering

A point $x$ in a Manifold $M$ is said to be nonwandering if, for every open Neighborhood $U$ of $x$, it is true that $\phi^{-n}U\cup U\not=\emptyset$ for a Map $\phi$ for some $n>0$. In other words, every point close to $x$ has some iterate under $\phi$ which is also close to $x$. The set of all nonwandering points is denoted $\Omega(\phi)$, which is known as the nonwandering set of $\phi$.

See also Anosov Diffeomorphism, Axiom A Diffeomorphism, Smale Horseshoe Map