New York Journal of Mathematics
Volume 14 (2008) 205-214


James T. Campbell and Roger L. Jones

Two-sided averages for which oscillation fails

Published: June 17, 2008
Keywords: ergodic theorem, square functions, Rohlin tower, oscillation
Subject: Primary 42B25; Secondary 40A30

We show that in any invertible, ergodic, measure-preserving system, the two-sided square function obtained by comparing forward averages with their backwards counterparts, will diverge if the (time) length of the averages grows too slowly. This contrasts with the one-sided case. We also show that for any sequence of times, certain weighted sums of the forward averages diverge. This contrasts with what would happen if the times increased rapidly and two-sided differences were considered.

Author information

James T. Campbell:
Department of Mathematical Sciences, Dunn Hall 373, University of Memphis, Memphis, TN 38152

Roger L. Jones:
Conserve School, 5400 N. Black Oak Lake Road, Land O'Lakes, WI 54540