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 [1], F can, more generally, be a compactly supported Borel automorphism preserving the

Lebesgue measure class.

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 of [GAMMA].

We reconsider the problem for area given by the

Lebesgue measure of the set (which includes the area considered in [1]) 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 on R.

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 of I.