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Things that could be mentioned, beyond the basic definitions:
K. E. Petersen (1970) "
A topologically strongly mixing symbolic minimal set" Trans. Amer. Math. Soc. 148 (1970), 603-612 We give here a "machinal" construction of a bilateral sequence with entries from 0, 1 whose orbit closure is topologically strongly mixing and minimal. We prove in addition that the flow we obtain has entropy zero, is uniquely ergodic, and fails to be measure-theoretically strongly mixing.
The above talks about blocks, and as far as I know, there are no Wikipedia articles on blocks (which are used all over the place in ergodic theory...)
67.198.37.16 (
talk)
06:40, 5 November 2020 (UTC)reply
Mixing stronger than ergodicity
The text says " Mixing asks for this ergodic property to hold between any two sets A and B, and not just between some set A and X."
It seems that the difference is not B vs X, but that mixing requires the non-empty intersection for all n, whereas ergodicity only requires it for some n (\forall vs \exist).
I'm not confident enough of this to edit the text. Could someone more familiar with this please check?
LachlanA (
talk)
11:09, 22 December 2022 (UTC)reply
This part is false as written : if you take A, B to be the singleton sets on two points in distinct orbits (which will exist as soon as X is uncountable) it will never occur that . I think that to define topological mixing you want A, B to be open.
On the other hand you certainly want "almost all n" and not "all n" in the definition : for a measurable transformation T of a standard Borel spaces, for an arbitrarily large N there will always exist nontrivial open sets A, B for which is empty for all .
This section "informal explanation" is a mess starting with the fourth paragraph. It should probably be pruned and the relevant information within incorporated in the rest of the article.
jraimbau (
talk)
13:26, 22 December 2022 (UTC)reply
Doesn't the total measure need to be = 1 ?
In the section Mixing in dynamical systems this passage appears:
Isn't it necessary to assume this is a probability measure space, that is, that µ(X) = 1 ?
Because then the measure of set A or B is the fraction of the total measure of X that A or B possesses, and then µ(A)µ(B) makes sense as the limit when the dynamical system is strongly mixing.