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 is isomorphic to the Cartesian product of an automorphism with a one-sided shift. The measure on the product space is always a nonsingular joining of the factor measures on each factor, but is not in general a product measure. It decomposes over the automorphic factor into a family of measures each of which is trivial on the tail σ-algebra of the shift. We give necessary and sufficient conditions under which the measure is a product measure and study the marginal measures in the general case. Necessary and sufficient conditions are given under which the marginal measure on the shift space is exact. We show that the theorem cannot be strengthened by constructing a finite measure-preserving ergodic two-to-one endomorphism whose product structure cannot be given product measure