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Invariant sets with zero measure and full Hausdorff dimension. (English) Zbl 0887.58042

Summary: For a subshift of finite type and a fixed Hölder continuous function, the zero measure invariant set of points where the Birkhoff averages do not exist is either empty or carries full Hausdorff dimension. Similar statements hold for conformal repellers and two-dimensional horseshoes, and the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously.

MSC:

37D99 Dynamical systems with hyperbolic behavior
37A99 Ergodic theory
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References:

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