Any minimal system on the circle is either uniquely ergodic or non-statistical

Abstract

This note establishes a connection between topological dynamics and statistical properties in ergodic theory. We begin by demonstrating that systems exhibiting the minimal oscillation property with respect to the Lebesgue measure m (that is, where the sequence of averages of the pullback of m accumulates on at least two distinct measures) are non-statistical. Consequently, this yields a sharp dichotomy: any minimal system on the circle is either uniquely ergodic or non-statistical with respect to the Lebesgue measure. As a result, we conclude that the set of minimal non-statistical systems with respect to Lebesgue measure is non-empty.