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On local metric pressure of dynamical systems

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Abstract

In this paper, a local version of the topological pressure of dynamical systems is presented. It is a function defined on the product space which does not depend on any measure. It is shown that, for any invariant measure, integration of the introduced function with respect to its corresponding diagonal measure results in the metric pressure of the dynamical system.

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References

  1. L. Breiman, The individual theorem of information theory. Ann. Math. Stat. 28, 809–811 (1957)

    Article  MathSciNet  Google Scholar 

  2. M. Brin, A. Katok, in On Local Entropy in Geometric Dynamics, Lecture Notes in Mathematics, Springer, New York, pp. 30–38 (1983)

  3. M. Carvalho, S.A.Pérez, A formula for the metric pressure. arXiv:1901.07198 (2019)

  4. W. Huang, Y. Yi, A local variational principle for pressure and its applications to equilibrium states. Isr. J. Math. 161(1), 29–74 (2007)

    Article  MathSciNet  Google Scholar 

  5. W. Huang, P. Lu, X. Ye, Measure-theoretical sensitivity and equicontinuity. Isr. J. Math. 183(1), 233–283 (2011)

    Article  MathSciNet  Google Scholar 

  6. P. Huang, C. Wang, Measure-theoretic pressure and topological pressure in mean metrics. Dyn. Syst. 34(2), 259–273 (2019)

    Article  MathSciNet  Google Scholar 

  7. D. Ma, M. Wu, Topological pressure and topological entropy of a semigroup of maps. Discrete Contin. Dyn. Syst. 31(2), 545–556 (2011)

    Article  MathSciNet  Google Scholar 

  8. B. McMillan, The basic theorems of information theory. Ann. Math. Stat. 24, 196–219 (1953)

    Article  MathSciNet  Google Scholar 

  9. Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv. 32, 54–114 (1977)

    Article  Google Scholar 

  10. Y. B. Pesin, B. Pitskel, Topological pressure and the variational principle for non-compact sets. Funct. Anal. Appl. 18(4), 307–318 (1984)

  11. R. Phelps, in Lectures on Choquet’s Theorem, Van Nostrand, Princeton (1966)

  12. M. Rahimi, A local approach to \(g\)-entropy. Kybernetika 51, 231–245 (2015)

    MathSciNet  MATH  Google Scholar 

  13. M. Rahimi, M. Hedyeloo, N. Bidabadi, A note on localization of entropy of doubly stochastic operators. Iran. J. Sci. Technol. Trans. Sci. 43, 2579–2584 (2019)

    Article  MathSciNet  Google Scholar 

  14. S. Shao, X. Ye, R. Zhang, Sensitivity and regionally proximal relation in minimal systems. Sci. China Ser. A Math. 51(6), 987–994 (2008)

    Article  MathSciNet  Google Scholar 

  15. D. J. Thompson, A thermodynamic definition of topological pressure for non-compact sets. Ergodic Theory Dyn. Syst. 31(2), 527–547 (2011)

  16. P. Walters, An Introduction to Ergodic Theory Springer, Berlin (1982)

  17. X. Ye, R.F. Zhang, On sensitive sets in topological dynamics. Nonlinearity 21(7), 1601–1620 (2008)

    Article  MathSciNet  Google Scholar 

  18. F.  Zeng, K. Yan, G. Zhang, Pre-image pressure and invariant measures. Ergodic. Theory Dyn. Syst. 27(3), 1037–1052 (2007)

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Acknowledgements

The authors would like to thank the referees for their comprehensive and useful comments which helped the improvement of this work to the present form.

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Rahimi, M., Assari, A. On local metric pressure of dynamical systems. Period Math Hung 82, 223–230 (2021). https://doi.org/10.1007/s10998-020-00355-w

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