Multiple disjointness and invariant measures on minimal distal flows

摘要

We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection {X-i}(i is an element of I) of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product Pi(i is an element of I) X-i is minimal if and only if Pi(i is an element of I) X-i(eq) is minimal, where X-i(eq) is the maximal equicontinuous factor of X-i. Most importantly, this result holds when each X-i is distal. When the phase group T is Z or R, we can apply this idea to construct large minimal distal product flows with many ergodic measures. We determine the exact cardinality of (ergodic) invariant measures on the universal minimal distal T-flow. Equivalently, we determine the cardinality of (extreme) invariant means on D(T), the space of distal functions on T. This cardinality is 2(c) for both ergodic and invariant measures. The size of the quotient of D(T) by a closed subspace with a unique invariant mean is found to be non-separable by using the same techniques.