Minimal Oscillation of Cesàro Averages Implies Non-Statisticality

We investigate the relationship between the global convergence of Cesàro averages and the pointwise statistical behavior of dynamical systems. First, we prove that if the Cesàro averages accumulate on at least two different measures (a property we call the minimal oscillation property) then the system is non-statistical. Second, we show that a system possesses a natural measure in the strong sense if and only if it is uniquely ergodic. As a consequence, every minimal homeomorphism on the circle possesses a natural measure in the strong sense which is physical and whose basin is the entire circle.