OK, I’ll keep this secret between you, me and the internet. :)

I like the dual formula; probably the analog is easier to interpret in spherical geometry.

]]>There is also a curious non-differential formula for volume one may glean from this approach. The area of a 2-dimensional cross-section is ∑(π-\theta) – 2π, so one gets a formula which is C ∑(π-\theta_e)l_e -2π Vol(P*), where P* is the dual polyhedron of all 2-planes that meet P, and C is some constant.

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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

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