Arkiv för Matematik

Irregular sets of two-sided Birkhoff averages and hyperbolic sets

Luis Barreira, Jinjun Li, and Claudia Valls

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Abstract

For two-sided topological Markov chains, we show that the set of points for which the two-sided Birkhoff averages of a continuous function diverge is residual. We also show that the set of points for which the Birkhoff averages have a given set of accumulation points other than a singleton is residual. A nontrivial consequence of our results is that the set of points for which the local entropies of an invariant measure on a locally maximal hyperbolic set does not exist is residual. This strongly contrasts to the Shannon–McMillan–Breiman theorem in the context of ergodic theory, which says that local entropies exist on a full measure set.

Article information

Source
Ark. Mat., Volume 54, Number 1 (2016), 13-30.

Dates
Received: 10 September 2013
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485802725

Digital Object Identifier
doi:10.1007/s11512-015-0214-2

Mathematical Reviews number (MathSciNet)
MR3475815

Zentralblatt MATH identifier
1361.37009

Rights
2015 © Institut Mittag-Leffler

Citation

Barreira, Luis; Li, Jinjun; Valls, Claudia. Irregular sets of two-sided Birkhoff averages and hyperbolic sets. Ark. Mat. 54 (2016), no. 1, 13--30. doi:10.1007/s11512-015-0214-2. https://projecteuclid.org/euclid.afm/1485802725


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References

  • Barreira, L., Dimension and Recurrence in Hyperbolic Dynamics, Progress in Mathematics 272, Birkhäuser, Basel, 2008.
  • Barreira, L., Ergodic Theory, Hyperbolic Dynamics and Dimension Theory, Universitext, Springer, Heidelberg, 2012.
  • Barreira, L. and Saussol, B., Multifractal analysis of hyperbolic flows, Comm. Math. Phys. 214 (2000), 339–371.
  • Barreira, L. and Schmeling, J., Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), 29–70.
  • Chen, E., Küpper, T. and Shu, L., Topological entropy for divergence points, Ergodic Theory Dynam. Systems 25 (2005), 1173–1208.
  • Fan, A.-H. and Feng, D.-J., On the distribution of long-term time averages on symbolic space, J. Stat. Phys. 99 (2000), 813–856.
  • Fan, A.-H., Feng, D.-J. and Wu, J., Recurrence, dimension and entropy, J. Lond. Math. Soc. (2) 64 (2001), 229–244.
  • Feng, D.-J., Lau, K.-S. and Wu, J., Ergodic limits on the conformal repellers, Adv. Math. 169 (2002), 58–91.
  • Li, J. and Wu, M., Divergence points in systems satisfying the specification property, Discrete Contin. Dyn. Syst. 33 (2013), 905–920.
  • Pesin, Ya. and Pitskel, B., Topological pressure and variational principle for non-compact sets, Funct. Anal. Appl. 18 (1984), 307–318.
  • Shereshevsky, M., A complement to Young’s theorem on measure dimension: the difference between lower and upper pointwise dimension, Nonlinearity 4 (1991), 15–25.
  • Thompson, D., The irregular set for maps with the specification property has full topological pressure, Dyn. Syst. 25 (2010), 25–51.