Tohoku Mathematical Journal

Irregular sets are residual

Luis Barreira, Jinjun Li, and Claudia Valls

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Abstract

For shifts with weak specification, we show that the set of points for which the Birkhoff averages of a continuous function diverge is residual. This includes topologically transitive topological Markov chains, sofic shifts and more generally shifts with specification. In addition, we show that the set of points for which the Birkhoff averages of a continuous function have a prescribed set of accumulation points is also residual. The proof consists of bridging together strings of sufficiently large length corresponding to a dense set of limits of Birkhoff averages. Finally, we consider intersections of finitely many irregular sets and show that they are again residual. As an application, we show that the set of points for which the Lyapunov exponents on a conformal repeller are not limits is residual.

Article information

Source
Tohoku Math. J. (2), Volume 66, Number 4 (2014), 471-489.

Dates
First available in Project Euclid: 21 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1432229192

Digital Object Identifier
doi:10.2748/tmj/1432229192

Mathematical Reviews number (MathSciNet)
MR3350279

Zentralblatt MATH identifier
1376.37032

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

Keywords
Irregular sets residual sets topological Markov chains

Citation

Barreira, Luis; Li, Jinjun; Valls, Claudia. Irregular sets are residual. Tohoku Math. J. (2) 66 (2014), no. 4, 471--489. doi:10.2748/tmj/1432229192. https://projecteuclid.org/euclid.tmj/1432229192


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References

  • S. Albeverio, M. Pratsiovytyi and G. Torbin, Topological and fractal properties of subsets of real numbers which are not normal, Bull. Sci. Math. 129 (2005), 615–630.
  • L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics 272, Birkhäuser, 2008.
  • L. Barreira, Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, Springer, 2012.
  • L. Barreira and B. Saussol, Multifractal analysis of hyperbolic flows, Comm. Math. Phys. 214 (2000), 339–371.
  • L. Barreira, B. Saussol and J. Schmeling, Distribution of frequencies of digits via multifractal analysis, J. Number Theory 97 (2002), 410–438.
  • L. Barreira and J. Schmeling, Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), 29–70.
  • R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470, Springer-Verlag, 1975.
  • E. Chen, T. Küpper and L. Shu, Topological entropy for divergence points, Ergodic Theory. Dynam. Systems 25 (2005), 1173–1208.
  • A.-H. Fan and D.-J. Feng, On the distribution of long-term time averages on symbolic space, J. Stat. Phys. 99 (2000), 813–856.
  • A.-H. Fan, D.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc. (2) 64 (2001), 229–244.
  • D.-J. Feng, K.-S. Lau and J. Wu, Ergodic limits on the conformal repellers, Adv. Math. 169 (2002), 58–91.
  • J. Li and M. Wu, Divergence points in systems satisfying the specification property, Discrete Contin. Dyn. Syst. 33 (2013), 905–920.
  • Ya. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen 18 (1984), 307–318.
  • M. Shereshevsky, A complement to Young's theorem on measure dimension: the difference between lower and upper pointwise dimension, Nonlinearity 4 (1991), 15–25.
  • D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst. 25 (2010), 25–51.