Editing Mixing (mathematics) (section)

== Topological mixing ==
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A form of mixing may be defined without appeal to a [[measure (mathematics)|measure]], using only the [[topology]] of the system.  A [[continuous map]] <math>f:X\to X</math> is said to be '''topologically transitive''' if, for every pair of non-empty [[open set]]s <math>A,B\subset X</math>, there exists an integer ''n'' such that
: <math>f^n(A) \cap B \ne \varnothing</math>
where <math>f^n</math> is the [[iterated function|''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 [[complete metric space]] with no [[isolated point]], then ''f'' is topologically transitive if and only if there exists a [[hypercyclic vector|hypercyclic point]] <math>x\in X</math>, that is, a point ''x'' such that its orbit <math>\{f^n(x): n\in \mathbb{N}\}</math> is [[dense set|dense]] in ''X''.

A system is said to be '''topologically mixing''' if, given open sets <math>A</math> and {{tmath|1= B }}, there exists an integer ''N'', such that, for all {{tmath|1= n>N }}, one has
: <math>f^n(A) \cap B \neq \varnothing.</math>

For a continuous-time system, <math>f^n</math> is replaced by the [[flow (mathematics)|flow]] {{tmath|1= \varphi_g }}, with ''g'' being the continuous parameter, with the requirement that a non-empty intersection hold for all {{tmath|1= \Vert g \Vert > N }}.

A '''weak topological mixing''' is one that has no non-constant [[continuous (topology)|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.