Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HISTORIC BEHAVIOR FOR FLOWS WITH THE GLUING ORBIT PROPERTY

  • Received : 2021.04.10
  • Accepted : 2021.12.06
  • Published : 2022.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the set of points with historic behavior (which is also called the irregular set) for continuous flows and suspension flows. In this paper under the hypothesis that (Xt)t is a continuous flow on a d-dimensional Riemaniann closed manifold M (d ≥ 2) with gluing orbit property, we prove that the set of points with historic behavior in a compact and invariant subset ∆ of M is either empty or is a Baire residual subset on ∆. We also prove that the set of points with historic behavior of a suspension flows over a homeomorphism satisfyng the gluing orbit property is either empty or Baire residual and carries full topological entropy.

Keywords

Acknowledgement

The author is deeply grateful to Paulo Varandas, Thiago Bomfim and Diego Daltro for the incentive and suggestions that helped to improve the structure of the paper.

References

  1. V. Araujo and V. Pinheiro, Abundance of wild historic behavior, Bull. Braz. Math. Soc. (N.S.) 52 (2021), no. 1, 41-76. https://doi.org/10.1007/s00574-019-00191-8
  2. L. Barreira, J. Li, and C. Valls, Irregular sets are residual, Tohoku Math. J. (2) 66 (2014), no. 4, 471-489. https://doi.org/10.2748/tmj/1432229192
  3. L. Barreira, J. Li, and C. Valls, Irregular sets for ratios of Birkhoff averages are residual, Publ. Mat. 58 (2014), suppl., 49-62. http://projecteuclid.org/euclid.pm/1400505225
  4. 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. https://doi.org/10.1007/BF02773211
  5. M. Bessa, M. J. Torres, and P. Varandas, On the periodic orbits, shadowing and strong transitivity of continuous flows, Nonlinear Anal. 175 (2018), 191-209. https://doi.org/10.1016/j.na.2018.06.002
  6. T. Bomfim, M. J. Torres, and P. Varandas, Topological features of flows with the reparametrized gluing orbit property, J. Differential Equations 262 (2017), no. 8, 4292- 4313. https://doi.org/10.1016/j.jde.2017.01.008
  7. T. Bomfim and P. Varandas, The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows, J. Differential Equations 267 (2019), no. 1, 228-266. https://doi.org/10.1016/j.jde.2019.01.010
  8. M. Carvalho and P. Varandas, Genericity of historic behavior for maps and flows, Nonlinearity 34 (2021), no. 10, 7030-7044. https://doi.org/10.1088/1361-6544/ac1f77
  9. J. Franks and M. Misiurewicz, Rotation sets of toral flows, Proc. Amer. Math. Soc. 109 (1990), no. 1, 243-249. https://doi.org/10.2307/2048385
  10. S. Kiriki, M.-C. Li, and T. Soma, Geometric Lorenz flows with historic behavior, Discrete Contin. Dyn. Syst. 36 (2016), no. 12, 7021-7028. https://doi.org/10.3934/dcds.2016105
  11. J. Li and M. Wu, Generic property of irregular sets in systems satisfying the specification property, Discrete Contin. Dyn. Syst. 34 (2014), no. 2, 635-645. https://doi.org/10.3934/dcds.2014.34.635
  12. H. Lima and P. Varandas, On the rotation sets of generic homeomorphisms on the torus 𝕋, Ergodic Theory Dynam. Systems 41 (2021), no. 10, 2983-3022. https://doi.org/10.1017/etds.2020.92
  13. M. Misiurewicz and K. Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490-506. https://doi.org/10.1112/jlms/s2-40.3.490
  14. Y. B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. https://doi.org/10.7208/chicago/9780226662237.001.0001
  15. Ya. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen. 18 (1984), no. 4, 50-63, 96. https://doi.org/10.1007/BF01076363
  16. D. Ruelle, Historical behaviour in smooth dynamical systems, in Global analysis of dynamical systems, 63-66, Inst. Phys., Bristol, 2001.
  17. F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity 21 (2008), no. 3, T33-T36. https://doi.org/10.1088/0951-7715/21/3/T02
  18. D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst. 25 (2010), no. 1, 25-51. https://doi.org/10.1080/14689360903156237