International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 47, Pages 2977-2988
doi:10.1155/S0161171203212217

Random entropy and recurrence

Karma Dajani1 and Ronald Meester2

1Department of Mathematics, Faculty of Mathematics and Computer Science, University of Utrecht, P.O. Box 80.010, Utrecht 3508 TA, The Netherlands
2Department of Mathematics, Faculty of Sciences, Vrije Universiteit, De Boelelaan 1081A, Amsterdam 1081 HV, The Netherlands

Received 19 December 2002

Copyright © 2003 Karma Dajani and Ronald Meester. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show that a cocycle, which is nothing but a generalized random walk with index set ℤd, with bounded step sizes is recurrent whenever its associated random entropy is zero, and transient whenever its associated random entropy is positive. This generalizes a well-known one-dimensional result and implies a Polya type dichotomy for this situation.