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Several authors have observed that heuristic probabilistic arguments suggest
that no divergent trajectories occur.
DIVERGENT TRAJECTORIES CONJECTURE.
The function
has no divergent trajectories,
i.e., there exists no integer
for which
If a divergent trajectory
exists, it
cannot be equidistributed
.
Indeed if one defines
then it can be proved that the condition (2.30) implies that
Theorem F constrains the possible behavior of divergent trajectories.
Indeed, associated to any divergent trajectory
is the
infinite set
and
for all
.
Since
for all
, Theorem F implies that
where
.
Roughly speaking, (2.32) asserts that the elements of a divergent
trajectory cannot go to infinity ``too slowly.''

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