Mixing (mathematics) - Topological Mixing

Topological Mixing

A form of mixing may be defined without appeal to a measure, only using the topology of the system. A continuous map is said to be topologically transitive if, for every pair of non-empty open sets, there exists an integer n such that

where is the n 'th iterate of f. In the operator theory, a topologically transitive bounded linear operator (a continuous linear map on a topological vector space) is usually called hypercyclic operator. A related idea is expressed by the wandering set.

Lemma: If X is a compact metric space, then f is topologically transitive if and only if there exists a hypercyclic point, that is, a point x such that its orbit is dense in X.

A system is said to be topologically mixing if, given sets and, there exists an integer N, such that, for all, one has

.

For a continuous-time system, is replaced by the flow, with g being the continuous parameter, with the requirement that a non-empty intersection hold for all .

A weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator.

Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.