One can look at two recent surveys by Crovisier: http://arxiv.org/pdf/1405.0305.pdf

Or Bonatti https://hal.archives-ouvertes.fr/hal-00463421/document or the longer treaty by Crovisier: http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.2896v1.pdf (Chapter 1) for how this concept is quite used in the study of smooth dynamics (from a C1-generic viewpoint).

Sure: I was certainly thinking of this construction in terms of limits of rational maps; modifying a rational map by multiplying by an “infinitesimal” (if you like, a “formal”) dipole at a point x produces a new rational map defined on a “cactus” obtained by attaching a “balloon” (i.e. a ) at x and its preimages by f, then attaching further 2nd order balloons to these 1st order balloons in the preimages of the first order balloons, etc. in an infinite tree; this is a sort of “algebraic limit” as the dipole size goes to zero, whose “geometric limit” is just f.

]]>But the most interesting foliations/laminations are those that are not transversely invariantly measured. Finite depth foliations can sometimes be described using more complicated weights (with values in non-Archimedean extensions of the reals) but the most interesting foliations admit no interesting invariant transverse measures at all.

For an introduction/reference, I recommend my book “Foliations and the geometry of 3-manifolds”, a pdf of which can be downloaded from the book page http://math.uchicago.edu/~dannyc/OUPbook/toc.html

]]>I was inspired by the following observation, perhaps due to Thurston: given a finite set of points as before, there exists a Kleinian group whose limit set is concentrated near this set of points: one can take the Schottky group generated by reflections in small circles centered at these points. ]]>

the order in which to do the approximation is (i) first choose a (big) N; then (ii) take small enough so that the Julia set is “predictable” (as a function of N).

In the more general situation that we start with an arbitrary (generic) f, as goes to zero the Julia sets of converge to . If we are taking f just to be then the points in are just the union of Y and the unit circle, and the th roots of points in Y, for all (positive) k. Thus, the absolute value of every such point is the th root of the absolute value of a point in Y; these absolute values go to 1 very quickly with N.

]]>Cheers,

K

Best, Pierre ]]>