topologically transitive

A continuous surjection f on a topological spaceMathworldPlanetmath X to itself is topologically transitive if for every pair of open sets U and V in X there is an integer n>0 such that fn(U)V, where fn denotes the n-th iterate of f.

If for every pair of open sets U and V there is an integer N such that fn(U)V for each n>N, we say that f is topologically mixing.

If X is a compactPlanetmathPlanetmath metric space, then f is topologically transitive if and only if there exists a point xX with a dense orbit, i.e. such that 𝒪(x,f)={fn(x):n} is dense in X.

Title topologically transitive
Canonical name TopologicallyTransitive
Date of creation 2013-03-22 13:41:05
Last modified on 2013-03-22 13:41:05
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Definition
Classification msc 37B99
Classification msc 54H20
Defines topologically mixing
Defines topological mixing