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Irregular sets for ratios of Birkhoff averages are residual
Barreira, Luis (Instituto Superior Técnico (Lisboa, Portugal). Departamento de Matemática)
Li, Jinjun (Minnan Normal University. School of Mathematics and Statistics)
Valls, Claudia (Instituto Superior Técnico (Lisboa, Portugal). Departamento de Matemática)

Data: 2014
Resum: It follows from Birkhoff's Ergodic Theorem that the irregular set of points for which the Birkhff 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 specfication 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.
Drets: Tots els drets reservats
Llengua: Anglès
Document: article ; recerca ; publishedVersion
Matèria: Birkho averages ; Irregular sets ; Weak specication
Publicat a: Publicacions matemàtiques, Vol. Extra, Núm. (2014) , p. 49-62, ISSN 0214-1493

DOI: 10.5565/PUBLMAT_Extra14_03

14 p, 317.2 KB

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