Publicacions Matemàtiques

Irregular sets for ratios of Birkhoff averages are residual

Luis Barreira, Jinjun Li, and Claudia Valls

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


It follows from Birkhoff's Ergodic Theorem that the irregular set of points for which the Birkhoff averages of a given continuous function diverge has zero measure with respect to any finite invariant measure. In strong contrast, for systems with the weak specification property, we show here that if the irregular set is nonempty, then it is residual. This includes topologically transitive topological Markov chains, sofic shifts and more generally shifts with the specification property. We consider also the more general case of ratios of Birkhoff averages of continuous functions and the case when the set of accumulation points of the ratios of Birkhoff averages is a prescribed closed interval. Finally, we give an application of our work to the pointwise dimension of a Gibbs measure on a repeller of a conformal map.

Article information

Publ. Mat., Volume EXTRA (2014), 49-62.

First available in Project Euclid: 19 May 2014

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]

Birkhoff averages irregular sets weak specification


Barreira, Luis; Li, Jinjun; Valls, Claudia. Irregular sets for ratios of Birkhoff averages are residual. Publ. Mat. EXTRA (2014), 49--62.

Export citation