An important issue in hyperbolic dynamics is that of absolute continuity. Suppose some neighbourhood of a smooth manifold is foliated by a collection of smooth submanifolds , where is some indexing set. (Here “smooth” may mean , or , or even more regularity depending on the context.)
Fixing a Riemannian metric on gives a notion of volume on the manifold , as well as a notion of “leaf volume” on each , which is just the volume form coming from the induced metric. Then one wants to understand the relationship between and .
The simplest example of this comes when is the (open) unit square and is the foliation into horizontal lines, so and . Then Fubini’s theorem says that if is any measurable set, then can be found by integrating the leaf volumes . In particular, if , then for almost every , and conversely, if has full measure, then for almost every .
For the sake of simplicity, let us continue to think of a foliation of the two-dimensional unit square by one-dimensional curves, and assume that these curves are graphs of smooth functions . The story in other dimensions is similar.
Writing gives a map , and we see that our foliation is the image under of the foliation by horizontal lines. Now we must make a very important point about regularity of foliations. The leaves of the foliation are smooth, and so depends smoothly on . However, no assumption has been made so far on the transverse direction — that is, dependence on . In particular, we cannot assume that depends smoothly on .
If depends smoothly on both and , then we have a smooth foliation, not just smooth leaves, and in this case basic results from calculus show that a version of Fubini’s theorem holds:
where the density functions can be determined in terms of the derivative of .
It turns out that when the foliation is by local stable and unstable manifolds of a hyperbolic map, smoothness of the foliation is too much to ask for, but it is still possible to find density functions such that Fubini’s theorem holds, and the relationship between -null sets and -null sets is as expected.
However, there are foliations where the leaves are smooth but the foliation as a whole does not even have the absolute continuity properties described in the previous paragraph. This includes a number of dynamically important examples, namely intermediate foliations (weak stable and unstable, centre directions in partial hyperbolicity) for certain systems. These take some time to set up and study. There is an elementary non-dynamical example first described by Katok; a similar example was described by Milnor (Math. Intelligencer 19(2), 1997), and it is this construction which I want to outline here, since it is elegant, does not require much machinery to understand, and neatly illustrates the possibility that absolute continuity may fail.
The example consists of two constructions: a set and a foliation of the square by a collection of smooth curves . These are built in such a way that has full Lebesgue measure () but intersects each curve in at most a single point, so that in particular for all .
1. The set
Given , consider the piecewise linear map given by
It is not hard to see that preserves Lebesgue measure and that is measure-theoretically conjugate to , where is the shift map on and is the -Bernoulli measure. The conjugacy is given by coding trajectories according to whether the th iterate falls into or .
Given , let be the number of times that lands in . Then by the strong law of large numbers, Lebesgue almost every has . Let be the set of all pairs such that this is true; then by Fubini’s theorem (applied to the foliation of the unit square into vertical lines). It only remains to construct a foliation that intersects in a “strange” way.
2. The foliation
The curves will be chosen so that two points and lie on the same if and only if they are coded by the same sequence in — that is, if and only if . Then because the limit of must be constant on if it exists, we deduce that each intersects the set at most once.
It remains to see that this condition defines smooth curves. Given , let be the binary expansion of , so that . Now let .
Fix , let be such that , and let , so that for all . Thus
For convenience of notation, write and . Then the relationship between and can be written as
This can be used to write in terms of the sequence . Indeed,
and so on, so that writing , one has
The summands are analytic functions of , and the sum converges uniformly on each interval , since for we have . In fact, this uniform convergence extends to complex values of in the disc of radius centred at , and so by the Weierstrass uniform convergence theorem, the function is analytic in .
As discussed above, is the graph of the analytic function , and each intersects at most once, despite the fact that has Lebesgue measure 1 in the unit square. This demonstrates the failure of absolute continuity.