Bounded ergodic constructions, disjointness, and weak limits of powers

Abstract

This paper is devoted to the disjointness property of powers of a totally ergodic bounded construction of rank 1 and some generalizations of this result. We look at applications to the problem when the Möbius function is independent of the sequence induced by a bounded construction. Interest in the subject matter of this paper is related to the following observation. Bounded constructions of rank 1, under the condition that all their nonzero powers are ergodic, have nontrivial weak limits of powers. This implies that the powers of the constructions are disjoint (in the sense of [1]) and, in view of the results in [2], this results in bounded constructions being independent of the Möbius function. Thus, the problem of disjointness of powers of transformations, which had previously been regarded by specialists as a problem within the framework of self-joining theory, has an interesting application. Sarnak’s well-known conjecture [3] states that a strictly ergodic homeomorphism S : X → X with zero topological entropy has the property