mixing action
Let be a topological space and let be a semigroup.
An action of on is (topologically) mixing
if, given any two open subsets , of ,
the intersection
is nonempty
for all except at most finitely many.
Example 1.
Let be a continuous function.
Then is topologically mixing if and only if
the action of the monoid on
defined by
is mixing according to the definition given above.
Example 2. Suppose is a discrete nonempty set and is a group; endow with the product topology. The action of on defined by
is mixing.
To prove this fact, we may suppose without loss of generality that and are two cylindric sets of the form:
for suitable finite subsets and functions .
Then the only chance for to be empty,
is that for some ,
such that :
but then, , 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 |