They cover Thurston maps, LattAaAeAeAes map quasiconformal and rough geometry, cell decompositions, expansion, Thurston maps with two or three postcritical points, visual metrics, symbolic dynamics, tile graphs, isotopies, subdivisions, quotients of Thurston maps, combinatorially expanding Thurston maps, invariant curves, the combinatorial expansion factor, the geometry of the visual sphere, relational Thurston maps and Lebesgue measure
, a combinatorial characterization of LattAaAeAeAes maps, and outlook and open problems.
As in , F can, more generally, be a compactly supported Borel automorphism preserving the Lebesgue measure
An analogous proof also works if we study the functions that are continuous except on a discrete set, a subset of a closed set with finite Lebesgue measure
or a bounded set.
assumed absolutely continuous with respect to a Lebesgue measure
, that describes the random behavior of the sequence.
To be more precise, let [lambda] x P denote the product of Lebesgue measure
and the probability measure on the Cartesian product I x [OMEGA], and let P denote the sigma algebra of [G.
for all [GAMMA] [subset or equal to] [0, 2[lambda]] with m([GAMMA]) [less than or equal to] [delta]([lambda]), where m([GAMMA]) is the Lebesgue measure
We reconsider the problem for area given by the Lebesgue measure
of the set (which includes the area considered in ) and for any given point in the plane.
Note that d(x, y) = [lambda](([phi](x) [union] [phi](y))\([phi](x) [intersection] [phi](y)), where [lambda] is Lebesgue measure
Remark 2: From Propositions 2 and 3, if [beta]([mu]) = [infinity] whenever [mu] is not equivalent to the Lebesgue measure
on (0,1], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
2]-integrable with respect to the Lebesgue measure
on [0, 2[pi]).
Appendices review partially ordered sets, Lebesgue measure
theory, and mollifications.
2], and let [omega] be an open polygonal convex subset of [OMEGA] and I [subset] [partial derivative][omega], with [absolute value of I] > 0; [absolute value of I] is the one-dimensional Lebesgue measure