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Journal of the London Mathematical Society 1991 s2-44(1):121-134; doi:10.1112/jlms/s2-44.1.121
© 1991 by London Mathematical Society
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© Oxford University Press

The Entropy of a Certain Infinitely Convolved Bernoulli Measure

J. C. Alexander and Don Zagier

Department of Mathematics, University of Maryland College Park, Maryland 20742, USA

An entropy was introduced by A. Garsia to study certain infinitely convolved Bernoulli measures (ICBMs)µß, and showed it was strictly less than 1 for ß the reciprocal of a Pisot-Vijayarghavan number. However, it is impossible to estimate values from Garsia's work. The first author and J. A. Yorke have shown this entropy is closely related to the ‘information dimension’ of the attractors of fat baker transformations Tß. When the entropy is strictly less than 1, the attractor is a type of strange attractor. In this paper, the entropy of µß is estimated for the case when ß = ø–1, where ø is the golden ratio. The estimate is fine enough to determine the entropy to several decimal places. The method of proof is totally unlike usual methods for determining dimensions of attractors; rather a relation with the Euclidean algorithm is exploited, and the proof has a number-theoretic flavour. It suggests that some interesting features of the Euclidean algorithm remain to be explored.


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