Title:An Effective Contraction Estimate in the Stable Subspaces of Phase Points in Hard Ball Systems

Abstract: 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.