mixing action

Let X be a topological spaceMathworldPlanetmath and let G be a semigroup. An action ϕ={ϕg}gG of G on X is (topologically) mixing if, given any two open subsets U, V of X, the intersectionDlmfMathworldPlanetmath Uϕg(V) is nonempty for all gG except at most finitely many.

Example 1. Let F:XX be a continuous functionMathworldPlanetmathPlanetmath. Then F is topologically mixing if and only if the action of the monoid on X defined by ϕn(x)=Fn(x) is mixing according to the definition given above.

Example 2. Suppose X is a discrete nonempty set and G is a group; endow XG with the product topology. The action of G on XG defined by


is mixing.

To prove this fact, we may suppose without loss of generality that U and V are two cylindric sets of the form:

U = {cXGc|E=u|E}
V = {cXGc|F=v|F}

for suitable finite subsets E,FG and functionsMathworldPlanetmath u,v:GX. Then the only chance for Uϕg(V) to be empty, is that e=gf for some eE, fF such that u(e)v(f): but then, gEF-1, which is finite.

Title mixing action
Canonical name MixingAction
Date of creation 2013-03-22 19:19:32
Last modified on 2013-03-22 19:19:32
Owner Ziosilvio (18733)
Last modified by Ziosilvio (18733)
Numerical id 5
Author Ziosilvio (18733)
Entry type Definition
Classification msc 37B05