WIAS Preprint No. 299, (1996)

Any set of irregular points has full Hausdorff dimension and full topological entropy


  • Barreira, Luis
  • Schmeling, Jörg
    ORCID: 0000-0001-6956-9463

2010 Mathematics Subject Classification

  • 58F15 58F11


  • Birkhoff averages, irregular points, local entropies, Lyapunov exponents, pointwise dimensions




We prove, for subshifts of finite type, conformal repellers, and two-dimensional horseshoes, that the set of points where both the pointwise dimension, local entropy, Lyapunov exponents, and Birkhoff averages do not exist carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute the first mathematical application of this theory.

Appeared in

  • Israel J. Math. 116 (2000), pp. 29--70.

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