The Entropy of a Certain Infinitely Convolved Bernoulli Measure
J. C. Alexander, Don Bernard Zagier · Journal of the London Mathematical Society · 1991
An entropy was introduced by A. Garsia to study certain infinitely convolved Bernoulli measures (ICBMs)μβ, 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 Tβ. When the entropy is strictly less than 1, the attractor is a type of strange attractor. In this paper, the entropy of μβ is estimated for the case when β = 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.