Oscillation in ergodic theory
In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.
Ergodic Theory and Dynamical Systems
Jones, R., Kaufman, R., Rosenblatt, J., & Wierdl, M. (1998). Oscillation in ergodic theory. Ergodic Theory and Dynamical Systems, 18 (4), 889-935. https://doi.org/10.1017/S0143385798108349