Open Access
2014 Irregular sets for ratios of Birkhoff averages are residual
Luis Barreira, Jinjun Li, Claudia Valls
Publ. Mat. 58(S1): 49-62 (2014).


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.


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Luis Barreira. Jinjun Li. Claudia Valls. "Irregular sets for ratios of Birkhoff averages are residual." Publ. Mat. 58 (S1) 49 - 62, 2014.


Published: 2014
First available in Project Euclid: 19 May 2014

zbMATH: 1309.37018
MathSciNet: MR3211826

Primary: 37B10

Keywords: Birkhoff averages , irregular sets , weak specification

Rights: Copyright © 2014 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.58 • No. S1 • 2014
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