REGULAR POINTS FOR ERGODIC SINAI MEASURES

Ergodic properties of smooth dynamical systems are considered. A point is called regular for an ergodic measure mu-if it is generic for mu-and the Lyapunov exponents at it coincide with those of mu. We show that an ergodic measure with no zero Lyapunov exponent is absolutely continuous with respect to unstable foliation [L] if and only if the set of all points which are regular for it has positive Lebesgue measure.