The asymptotically additive topological pressure: variational principle for non-compact and intersection of irregular sets

ABSTRACT

Let $(X,d,f)$ be a dynamical system, where $(X,d)$ is a compact metric space and $f:X\to X$ is a continuous map. Using the concepts of g-almost product property and uniform separation property introduced by Pfister and Sullivan in Pfister and Sullivan [On the topological entropy of saturated sets, Ergodic Theory Dyn. Syst. 27 (2007), pp. 929–956], we give a variational principle for certain non-compact with relation to the asymptotically additive topological pressure. We also study the set of points that are irregular for a collection finite or infinite of asymptotically additive sequences and we show that carried the full asymptotically additive topological pressure. These results are suitable for systems such as mixing shifts of finite type, β-shifts, repellers and uniformly hyperbolic diffeomorphisms.

KEYWORDS:

• Asymptotically additive topological pressure
• variational principle
• sequentially saturated
• specification property
• β-shifts
• hyperbolic systems
• nonconformal repellers
• cocycles under shift of the finite type

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 37C50
• 37C45
• 37B10
• 37D20

Acknowledgments

The author is grateful to P. Varandas and V. Ramos for providing a preliminary version of this paper and making important suggestions that helped improve the presentation of the text. The author is also grateful to anonymous referees for giving us a simpler proof for the Theorem A and suggestions that helped to improve the text.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by a Instituto Nacional de Ciência e Tecnologia de Matemática (INCT-Mat)/Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes)-Brazil postdoctoral fellowship at University of Bahia.