Apparent violation of Fubini's theorem
Fubini's nightmare is a seeming violation of Fubini's theorem, where a nice space, such as the square formula_1 is foliated by smooth fibers, but there exists a set of positive measure whose intersection with each fiber is singular (at most a single point in Katok's example). There is no real contradiction to Fubini's theorem because despite smoothness of the fibers, the foliation is not absolutely continuous, and neither are the conditional measures on fibers.
Existence of Fubini's nightmare complicates fiber-wise proofs for center foliations of partially hyperbolic dynamical systems: these foliations are typically Hölder but not absolutely continuous.
A hands-on example of Fubuni's nightmare was suggested by Anatole Katok and published by John Milnor.
A dynamical version for center foliation was constructed by Amie Wilkinson and Michael Shub.
Katok's construction.
Foliation.
For a formula_2 consider the coding of points of the interval formula_3 by sequences of zeros and ones, similar to the binary coding, but splitting the intervals in the ratio formula_4. (As for the binary coding, we identify formula_5 with formula_6)
The point, corresponding to a sequence formula_7 is given explicitly by
 formula_8
where
formula_9
is the length of the interval after first formula_10 splits.
For a fixed sequence formula_11 the map formula_12 is analytic. This follows from the Weierstrass M-test: the series for formula_12 converges uniformly on compact subsets of the intersection formula_14 In particular, formula_15 is an analytic curve.
Now, the square formula_16 is foliated by analytic curves formula_17
Set.
For a fixed formula_18 and random formula_19 sampled according to the Lebesgue measure, the coding digits formula_20 are independent Bernoulli random variables with parameter formula_18, namely formula_22 and formula_23
By the law of large numbers, for each formula_18 and almost every formula_25
 formula_26
By Fubini's theorem, the set
 formula_27
has full Lebesgue measure in the square formula_16.
However, for each fixed sequence formula_29 the limit of its Cesàro averages formula_30 is unique, if it exists. Thus every curve formula_31 either does not intersect formula_32 at all (if there is no limit), or intersects it at the single point formula_33 where
 formula_34
Therefore, for the above foliation and set formula_32, we observe a Fubini's nightmare.
Wilkinson–Shub construction.
Wilkinson and Shub considered diffeomorphisms which are small perturbations of the diffeomorphism formula_36 of the three dimensional torus formula_37 where formula_38 is the Arnold's cat map. This map and its small perturbations are partially hyperbolic. Moreover, the center fibers of the perturbed maps are smooth circles, close to those for the original map.
The Wilkinson and Shub perturbation is designed to preserve the Lebesgue measure and to make the diffeomorphism ergodic with the central Lyapunov exponent formula_39 Suppose that formula_40 is positive (otherwise invert the map). Then the set of points, for which the central Lyapunov exponent is positive, has full Lebesgue measure in formula_41
On the other hand, the length of the circles of the central foliation is bounded above. Therefore, on each circle, the set of points with positive central Lyapunov exponent has to have zero measure. More delicate arguments show that this set is finite, and we have the Fubini's nightmare.
References.
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