 
Summary: MOSHER'S ARATIONALITY CRITERION
In Lee Mosher's talk at U. of Chicago on 2/27/03, he gave an example
demonstrating how to tell if a lamination is arational. One way to
define arational is that the lamination meets every simple closed curve
essentially. On the torus, these correspond to measured laminations
of irrational slope, so are a generalization of irrational numbers. A
lamination may be specified by a sequence of splittings of train tracks,
where each train track is some coarse view of the lamination, where
one can resolve distinct leaves only up to a certain scale, after which
leaves merge together at branches, called cusps. Mosher discussed a
particular kind of sequence of splittings, where one has a distinguished
cusp, marked by . One splits the train track sequentially at the cusp
, and each splitting is specified by an L or R, depending on whether
the branch splits to the left or right (see figure 1). When the two cusps
agree, there is only one way to split, so we do not need to record it.
If the train track fully carries a lamination (meaning that there is a
smooth homotopy of the lamination into the train track), then this
sequence of splittings is uniquely determined by the lamination. Thus,
a sequence LRLLRRRL... defines a unique sequence of splittings of the
train track along the distinguished cusp.
