  We put forward a conjecture of recurrence for a gas of hard spheres that
collide elastically in a finite volume. The dynamics consists of a sequence of
instantaneous binary collisions. We study how the numbers of collisions of
different pairs of particles grow as functions of time. We observe that these
numbers can be represented as a time-integral of a function on the phase space.
Assuming the results of the ergodic theory apply, we describe the evolution of
the numbers by an effective Langevin dynamics. We use the facts that hold for
these dynamics with probability one, in order to establish properties of a
single trajectory of the system. We find that for any triplet of particles
there will be an infinite sequence of moments of time, when the numbers of
collisions of all three different pairs of the triplet will be equal. Moreover,
any value of difference of collision numbers of pairs in the triplet will
repeat indefinitely. On the other hand, for larger number of pairs there is but
a finite number of repetitions. Thus the ergodic theory produces a limitation
on the dynamics.
