Talk:Mixing (mathematics)

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Things to mention[edit]

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[edit]

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

A\cap T^{-n}B\not =\emptyset

. 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

A\cap T^{-n}B

is empty for all

0\leq n\leq N

.

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]