The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Revista Matemática Iberoamericana

Full-Text PDF (312 KB) | Metadata | Table of Contents | RMI summary
Online access to the full text of Revista Matemática Iberoamericana is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at
Volume 34, Issue 2, 2018, pp. 853–878
DOI: 10.4171/RMI/1006

Published online: 2018-05-28

Topological entropy of irregular sets

Luis Barreira[1], Jinjun Li[2] and Claudia Valls[3]

(1) Instituto Superior Técnico, Lisboa, Portugal
(2) Guangzhou University, China
(3) Instituto Superior Técnico, Lisboa, Portugal

For expansive continuous maps with the specification property, we compute the topological entropy of the irregular set for the Birkhoff averages of a continuous function. This is the set of points for which the Birkhoff averages do not converge. The entropy is expressed in terms of a conditional variational principle. We also consider the general case of irregular sets obtained from ratios of Birkhoff averages of continuous functions. Moreover, we obtain a conditional variational principle for the topological entropy of the family of subsets of the irregular set formed by the points such that the set of accumulation points of the ratio of Birkhoff averages is a given interval. As nontrivial applications, we obtain conditional variational principles for the topological entropy of the level sets of local entropies, pointwise dimensions and Lyapunov exponents both on repellers and hyperbolic sets.

Keywords: Irregular sets, topological entropy, specification property

Barreira Luis, Li Jinjun, Valls Claudia: Topological entropy of irregular sets. Rev. Mat. Iberoam. 34 (2018), 853-878. doi: 10.4171/RMI/1006