Jim Olsen
North Dakota State University

Title: Generalizations of Bourgain's Return Times Theorem

The return times theorem of Bourgain says that if $\tau $ is an ergodic measure preserving transformation of the probability space $(X, \Cal F, \mu )$, $A \in \Cal F $, then for a.e. $x \in X$, the sequence $\{a_k(x) = \displaystyle {\Cal X_A(x)} \}$ is a universally good sequence for the pointwise ergodic theorem, i.e., that if $(Y, \Cal G , \nu)$ is another probability space and $\sigma $ is a measure preserving point transformation of $Y$, then the sequence $\{\displaystyle {\frac 1N \sum _{k=0}^{N-1}{a_k(x) T^kf(y)}}\}$ converges a.e. $\nu $ for all $f \in L_{\infty}(Y)$, where the linear operator $T$ is defined by $Tf(y) = f \circ \sigma (y)$. The work of several authors then shows that this sequence will converge a.e. $\nu $ if instead $T$ is a Dunford-Schwartz operator(i.e., simultaneously a contraction of $L_1(Y) $and $L_{\infty}(Y)$ and $f\in L_P(Y)$, $1\leq p \leq \infty $, or if $T$ is a positively dominated contraction of $L_p(Y)$, $p$ fixed, $1 < p < \infty $, and $f \in L_p(Y)$, as well as other cases.

We will consider the more general situation of $\{a_k(x) = S^kg(x)\} $, where $S$ is a Dunford-Schwartrz operator of $L_p(X)$. In this setting we obtain convergence for Dunford-Schwartz operators and $f \in L_p(Y)$ for all $p$, $1\leq p \leq \infty $ and for positively dominated contractions for $f \in L_p(Y)$, $p $ fixed, $1 < p < \infty $ when $g$ is bounded. When $f \in L_p(X) $, $1 \leq p \leq \infty $, we obtain convergence for $S$ a Dunford-Schwartz operator and $g \in L_q(Y)$, where $\displaystyle {\frac 1p + \frac 1q =1} $, and for $g \in L_r(Y)$ when $r > q$ and $S$ a positively dominated contraction of $L_r(Y)$.