In this paper we prove the following result, useful and often needed in the
study of the ergodic properties of hard ball systems: In any such system, for
any phase point x with a non-singular forward trajectory and infinitely many
connected collision graphs on that forward orbit, it is true that for any small
number epsilon there is a stable tangent vector w of x and a large enough time
t>>1 so that the vector w undergoes a contraction by a factor of less than
epsilon in time t. Of course, the Multiplicative Ergodic Theorem of Oseledets
provides a much stronger conclusion, but at the expense of an unspecified
zero-measured exceptional set of phase points, and this is not sufficient in
the sophisticated studies the ergodic properties of such flows. Here the
exceptional set of phase points is a dynamically characterized set, so that it
suffices for the proofs showing how global ergodicity follows from the
localone.