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^nU intersection U!=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

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Nonwandering." From MathWorld--A Wolfram Web Resource.

Subject classifications