Given a map
and
an initial argument
,
we can iterate the map to get a finite forward orbit modulo a prime
. In particular, for
a quadratic map
,
where
is
constant, work by Pollard suggests that the forward orbit should have length on the
order of
.
We give a heuristic argument that suggests that the statistical properties of
this orbit might be very similar to the birthday problem random variable
, for an
day year, and
offer considerable experimental evidence that the limiting distribution of the orbit lengths,
divided by
,
for
as
, converges to the
limiting distribution of
,
as
.
Keywords
arithmetic dynamics, birthday problem, forward orbit modulo
$p$, random maps
