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## Invariant sets with zero measure and full Hausdorff
dimension

### Luis Barreira and Jörg Schmeling

**Abstract.**
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.

*Copyright 1997 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**03** (1997), pp. 114-118
- Publisher Identifier: S 1079-6762(97)00035-8
- 1991
*Mathematics Subject Classification*. Primary 58F15,
58F11
- Received by the editors September 2, 1997
- Posted on October 29, 1997
- Communicated by Svetlana Katok
- Comments (When Available)

**Luis Barreira**

Departamento de Matemática, Instituto Superior Técnico, 1096
Lisboa, Portugal

*E-mail address:* `barreira@math.ist.utl.pt`

*URL:* `http://www.math.ist.utl.pt/~barreira/`

**Jörg Schmeling**

Fachbereich Mathematik und Informatik, Freie Universität Berlin,
Arnimallee 2-6, D-14195 Berlin, Germany

*E-mail address:* `shmeling@math.fu-berlin.de`

Luis Barreira was partially supported by the projects PRAXIS XXI,
2/2.1/MAT/199/94 and JNICT,
PBIC/C/MAT/2140/95. Jörg Schmeling was supported by the
Leopoldina-Forderpreis.

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