Ergodic Theory and Dynamical Systems

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

a1 UPR 9016 CNRS, Institut de Mathématiques de Luminy, 163 avenue de Luminy, case 907, 13288 Marseille, cedex 9, France (e-mail:
a2 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK (e-mail:


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)

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