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Oscillation in ergodic theory

Published online by Cambridge University Press:  01 August 1998

ROGER L. JONES ,
ROBERT KAUFMAN ,
JOSEPH M. ROSENBLATT  and
MÁTÉ WIERDL
ROGER L. JONES
Affiliation:
Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago, IL 60614, USA (e-mail: rjones@condor.depaul.edu)
ROBERT KAUFMAN
Affiliation:
Department of Mathematics, University of Illinois at Urbana, Urbana, IL 61801, USA (e-mail: jrsnbltt@symcom.math.uiuc.edu)
JOSEPH M. ROSENBLATT
Affiliation:
Department of Mathematics, University of Illinois at Urbana, Urbana, IL 61801, USA (e-mail: jrsnbltt@symcom.math.uiuc.edu)
MÁTÉ WIERDL
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (e-mail: wierdlm@mathsci.msci.memphis.edu)
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Abstract

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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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 4 , August 1998 , pp. 889 - 935
Copyright
© 1998 Cambridge University Press