In this talk I will describe various conditions that imply a sequence of operators diverge. These conditions will be used to show ergodic averages along lacunary sequences, convolution powers of certain measures, and other kinds of averages can diverge. A discussion of Bourgain's entropy method will be included.
In this talk I will discuss what is known about ergodic averages obtained by taking convolution powers of a fixed measure. The $L^2$ result will be obtained, and used to obtain the weak type (1,1) result. It will be shown how to obtain a ``good - $\lambda $'' inequality between the maximal operator associated with this sequence of operators, and the usual ergodic maximal function. Some open problems will be mentioned.
In this talk I will look at the oscillation of the sequence of classical ergodic averages, and establish norm inequalities for the oscillation operators. Some of the results will be extended to other settings, and applications will be discussed.