Ergodic Theory and Dynamical Systems



Computing the dimension of dynamically defined sets: E_2 and bounded continued fractions


OLIVER JENKINSON a1p1 and MARK POLLICOTT a2
a1 UPR 9016 CNRS, Institut de Mathématiques de Luminy, 163 avenue de Luminy, case 907, 13288 Marseille, cedex 9, France (e-mail: omj@maths.qmw.ac.uk)
a2 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK (e-mail: mp@ma.man.ac.uk)

Abstract

We present a powerful approach to computing the Hausdorff dimension of certain conformally self-similar sets. We illustrate this method for the dimension \mathop{\rm dim}\nolimits_H(E_2) of the set E_2, consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking feature of this method is that the successive approximations converge to \mathop{\rm dim}\nolimits_H(E_2) at a super-exponential rate

(Received November 11 1999)
(Revised September 29 2000)


Correspondence:
p1 Present address: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK