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I was reminded of this topic by some of the answers to this question, where it was noted that "typical" measure-preserving transformations are weak-mixing but not strong-mixing for several senses of "typical". As a result, it occurred to me that I do not know of any very natural, explicit examples of transformations which are weakly but not strongly mixing. So,

What are some good examples of measure-preserving transformations which are weak-mixing but not strong-mixing?

To clarify "good": I'm particularly interested in examples where it can be proved in a concise and self-contained manner that weak mixing occurs and strong mixing does not, in examples which arise constructively, and in examples which arise directly from a continuous transformation of a compact metric space (as opposed to abstract measure-theoretic constructions).

Thanks in advance!

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6 Answers 6

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The Chacon transformation has a nice and fairly explicit description as a uniquely ergodic subshift: set $B_0=0$ and set $B_{k+1}=B_kB_k1B_k$.

The subshift is the set of all infinite words, all of whose finite subwords occur as a someword of some $B_k$.

From this it is easy to see why the lengths $n=|B_k|$ fail to be good times for strong mixing. As mentioned previously Parry shows the weak mixingness.

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I think a good example (which may not qualify as consise, but does as self contained) is given in the famous paper of Anosov and Katok. There they construct smooth diffeomorphisms of the disc being weak mixing (and many other ergodic properties) as limits of diffeomorphisms conjugated to rational rotations (which in the limit have "Liouville" rotation number in the anulus). Since they are limit of conjugations to rotations, one can prove that there is a sequence $f^{q_n} \to Id$ and thus it cannot be strongly mixing (a more modern aproach can be found in this paper and this one).

Fayad has also studied this problem for reparametrizations of linear flows of tori.

Other example, quite more involved (not at all consise nor self contained) is Interval Exchange transformations, it is proved by Katok that they are not Strongly Mixing, and recently, Avila and Forni have proved that they are weak mixing.

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Oddly, I was just looking for such a transformation last week, and ran across various "cutting-and-stacking" constructions, of which this paper by Chacon is a short, self-contained example. (This probably meets your first two conditions, but isn't a continuous transformation.)

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I think that the examples in "K. Frączek, M. Lemańczyk, On mild mixing of special flows over irrational rotations under piecewise smooth functions" satisfy your requirements although they are flows and not transformations.

The book of Steve Alpern and Raj Prasad has a proof that whatever is typical in the space of measure preserving transformations (of a compact metric space) is also typical in the space of homeomorphisms of that space.

Kelly Yancey's has several papers about constructions of rigid (hence not mixing) and weakly mixing transformations on the two torus and the Klein bottle.

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This kind of transformations occur in (papers on) Statistical Mechanics (general Ising models).

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Just consider a special flow above a circle rotation of irrational angle and under a ceiling function {x}+1

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    $\begingroup$ Can you please add some details to your answer? I find it hard to understand what you mean. Thank you. $\endgroup$
    – Ricardo Andrade
    Commented Sep 23, 2014 at 13:47

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