Communications in Mathematical Physics

, Volume 207, Issue 1, pp 145–171

Multifractal Analysis of Lyapunov Exponent for Continued Fraction and Manneville–Pomeau Transformations and Applications to Diophantine Approximation

  • Mark Pollicott
  • Howard Weiss

DOI: 10.1007/s002200050722

Cite this article as:
Pollicott, M. & Weiss, H. Comm Math Phys (1999) 207: 145. doi:10.1007/s002200050722


We extend some of the theory of multifractal analysis for conformal expanding systems to two new cases: The non-uniformly hyperbolic example of the Manneville–Pomeau equation and the continued fraction transformation. A common point in the analysis is the use of thermodynamic formalism for transformations with infinitely many branches.

We effect a complete multifractal analysis of the Lyapunov exponent for the continued fraction transformation and as a consequence obtain some new results on the precise exponential speed of convergence of the continued fraction algorithm. This analysis also provides new quantitative information about cuspital excursions on the modular surface.

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Mark Pollicott
    • 1
  • Howard Weiss
    • 2
  1. 1.Department of Mathematics, The University of Manchester, Oxford Road, M13 9PL, Manchester, UK. E-mail:
  2. 2.Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.¶E-mail: weiss@math.psu.eduUS