ROHLIN FACTORS, PRODUCT FACTORS, AND JOININGS FOR N-TO-ONE MAPS

In this paper we show that every nonsingular conservative ergodic n-to -one endomorphism of a Lebesgue probability space has a factor which i s isomorphic to the Cartesian product of an automorphism with a one-si ded shift. The measure on the product space is always a nonsingular jo ining of the factor measures on each factor, but is not in general a p roduct measure. It decomposes over the automorphic factor into a famil y of measures each of which is trivial on the tail sigma-algebra of th e shift. We give necessary and sufficient conditions under which the m easure is a product measure and study the marginal measures in the gen eral case. Necessary and sufficient conditions are given under which t he marginal measure on the shift space is exact. We show that the theo rem cannot be strengthened by constructing a finite measure-preserving ergodic two-to-one endomorphism whose product structure cannot be giv en product measure.