Canad. J. Math. Vol. 51 (5), 1999 pp. 996–1019 Counting in Ergodic Theory Roger L. Jones, Joseph M. Rosenblatt and M´ at´ e Wierdl Abstract. Many aspects of the behavior of averages in ergodic theory are a matter of counting the number of times a particular event occurs. This theme is pursued in this article where we consider large deviations, square functions, jump inequalities and related topics. Received by the editors April 2, 1998; revised February 19, 1999 and April 16,1999. R. Jones is partially supported by NSF Grant DMS—9531526. J. Rosenblatt is partially supported by NSF Grant DMS—9705228. M. Wierdl is partially supported by NSF Grant DMS—9500577. AMS subject classiﬁcation: ****MISSING AMS CLASSIFICATION****. c Canadian Mathematical Society 1999. 996