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

Published online by Cambridge University Press:  02 October 2001

OLIVER JENKINSON  and
MARK POLLICOTT
OLIVER JENKINSON
Affiliation:
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) Present address: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
MARK POLLICOTT
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK (e-mail: mp@ma.man.ac.uk)
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Abstract

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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

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 21 , Issue 5 , October 2001 , pp. 1429 - 1445
Copyright
2001 Cambridge University Press