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Abstract:
An entropy was introduced by A. Garsia to study certain infinitely convolved Bernoulli measures (ICBMs) μ\sbβ, and showed it was strictly less than 1 for β 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 T\sbβ. When the entropy is strictly less than 1, the attractor is a type of strange attractor. In this paper, the entropy of μ\sbβ is estimated for the case β =φ\sp-1, where φ 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 some interesting features of the Euclidean algorithm remain to be explored.
