# Item

ITEM ACTIONSEXPORT

**The entropy of a certain infinitely convolved Bernoulli measure.**

Alexander, J., & Zagier, D. (1991). The entropy of a certain infinitely convolved
Bernoulli measure.* Journal of the London Mathematical Society. Second Series,* *44*(1),
121-134.

### Basic

show hide**Item Permalink:**https://hdl.handle.net/21.11116/0000-0004-3927-1

**Version Permalink:**https://hdl.handle.net/21.11116/0000-0004-3928-0

**Genre:**Journal Article

### Files

show Files### Locators

show### Content

show
hide

**Free keywords:**-

**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.

### Details

show
hide
Date issued: 1991

Other: 111

**Language(s):**

**Dates:**

**Publication Status:**Issued

**Pages:**-

**Publishing info:**-

**Table of Contents:**-

**Rev. Type:**-

**Identifiers:**eDoc: 744931

Other: 111

**Degree:**-

### Event

show### Legal Case

show### Project information

show### Source 1

show
hide

**Title:**Journal of the London Mathematical Society. Second Series

**Source Genre:**Journal

^{ }Creator(s):**Affiliations:**

**Publ. Info:**London Mathematical Society, London; Oxford University Press, Oxford

**Pages:**-**Volume / Issue:**44 (1)**Sequence Number:**-**Start / End Page:**121 - 134**Identifier:**ISSN: 0024-6107