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Entropy for random group actions

Published online by Cambridge University Press:  01 February 1998

ROBERT BURTON ,
KARMA DAJANI  and
RONALD MEESTER
ROBERT BURTON
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA (e-mail: [email protected])
KARMA DAJANI
Affiliation:
Department of Mathematics, University of Utrecht, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands (e-mail: {dajani, meester}@math.ruu.nl)
RONALD MEESTER
Affiliation:
Department of Mathematics, University of Utrecht, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands (e-mail: {dajani, meester}@math.ruu.nl)
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Abstract

We consider the entropy of systems of random transformations, where the transformations are chosen from a set of generators of a ${\Bbb Z}^d $ action. We show that the classical definition gives unsatisfactory entropy results in the higher-dimensional case, i.e. when $d \geq 2$. We propose a definition of the entropy for random group actions which agrees with the classical definition in the one-dimensional case, and which gives satisfactory results in higher dimensions. This definition is based on the fibre entropy of a certain skew product. We identify the entropy by an explicit formula which makes it possible to compute the entropy in certain cases.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 1 , February 1998 , pp. 109 - 124
Copyright
1998 Cambridge University Press

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