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What is a simple example of a (continuous) foliation of a manifold that is not absolutely continuous?

(A foliation is said to be absolutely continuous if holonomy maps between smooth transversals send zero volume sets to zero volume sets.)

There's an example in Section 10.2 of Hasselblatt-Katok's Handbook of dynamical systems volume 1B, but it involves thinking about entropy. I wonder if there's something simpler, perhaps built from a non-absolutely continuous function, such as Cantor's devil staircase.

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    $\begingroup$ Since a) all (topological) Cantor sets embedded in ℝ are equivalent to each other by a homeomorphism of ℝ, and b) a Cantor set in ℝ can have either 0 or positive Lebesgue measure, it follows that one can construct a (topological) foliation F of the torus by curves such that F is not absolutely continuous. $\endgroup$
    – Daniel Asimov
    Commented Sep 9, 2023 at 19:58
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    $\begingroup$ Indeed, one can accomplish this, but the foliation in Hasselblatt-Katok, and the one in Milnor's article, exhibit an even more pathological property; see the edit to my answer. $\endgroup$
    – Vaughn Climenhaga
    Commented Sep 10, 2023 at 0:39
  • $\begingroup$ @DanielAsimov Thanks, this is exactly the type of thing I was hoping for, should have thought of it myself! And, it's nice that the individual leaves are lines, as nice a curve as one could hope for. $\endgroup$
    – RegularGraph
    Commented Sep 10, 2023 at 17:54

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I don’t have the Handbook near me at the moment so I can’t look at the example you mention, but at https://vaughnclimenhaga.wordpress.com/2013/11/24/fubini-foiled/ I describe a construction of Milnor’s (see the Intelligencer article linked to there) that is my go-to example for an elementary instance of this phenomenon. It’s similar to Katok’s original example but not identical and avoids things like entropy.

EDIT: This answer by Pengfei to a related MO question gives a very simple example of a continuous foliation which has a holonomy map that is not absolutely continuous, essentially following the idea that Daniel Asimov described in the comments (put the middle-third Cantor set on one side of a square, a Cantor set of positive Lebesgue measure on the other side, and connect corresponding points by straight lines). However, this foliation does have the weaker absolute continuity property that if $A\subset [0,1]^2$ has positive (2-dimensional) Lebesgue measure, then for many leaves $W$ of the foliation, $A\cap W$ has positive (1-dimensional) Lebesgue measure. (Here "many leaves" means "a set of leaves whose union has positive 2-dimensional Lebesgue measure.) The foliation described in Milnor's article (and my blog post) fails to have even this weaker property.

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  • $\begingroup$ Thanks! It is nice that one can have a more pathological property, I was kind of hoping for that. Milnor's example has the property that there is a full measure set that intersects each leaf of the foliation at most once! $\endgroup$
    – RegularGraph
    Commented Sep 10, 2023 at 17:57

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