On pressure of dynamical systems induced by probability bi-sequences

Articles in Press

Document Type : Original Article

Authors

Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran

10.22098/jfga.2026.19659.1208

Abstract

In this paper, we introduce a new family of metrics on topological dynamical systems, induced by probability bi-sequences, which generalizes both the classical Bowen metric and the mean metric. Using these metrics, we define measure-theoretic and topological pressure and show that these quantities coincide with the classical topological pressure and the sum of measure-theoretic entropy and integral of the potential function, respectively. The results hold under the following mild condition on the probability bi-sequence Γ ={γm,n}m,n≥ 0:

limsupn→∞ [(γn*)-1/n] < ∞,

where γn* := min0≤ i ≤ nγi,n. This condition is satisfied for a broad class of bi-sequences, including the uniform weights γm,n = 1/(n+1) that recover the mean metric case. As an application, we extend the pressure versions of Katok's entropy formula to this more general setting. Our work unifies and generalizes several previous results on pressure in mean metrics.

Keywords

References

 1. R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer.Math. Soc., 114 (1965), 309-319.
2. M. Carvalho, S. A. P´erez, A formula for the metric pressure, arXiv.org. math.arXiv:1901.07198. (2019).
3. W. Cheng, Y. Zhao, Y. Cao, Pressures for asymptotically sub-additive potentials under a mistake function, Discrete Contin. Dyn. Syst. Ser. A 32(2) (2013), 487-497.
4. M. Gr¨oger and T. J¨ager, Some remarks on modified power entropy, Dynamics and Numbers, Contem. Math., 669(2016), 105-122.
5. L. He, J. Lv, L. Zhou, Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica, Engl. Ser. 20 (4) (2004), 709-718.
6. P. Huang, E. Chen, C. Wang, Entropy formulae of conditional entropy in mean metrics, Discrete and Continuous Dynamical Systems, 38, No. 10 (2018), 5129-5144.
7. W. Huang, Z. Wang, X. Ye, Measure complexity and M¨obius disjointness, Advances in Mathematics, 347 (2019), 827-858.
8. P. Huang, C. Wang, Measure-theoretic and topological pressure in mean metrics, Dynamical Systems, 34 (2019), 259-273.
9. A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ.Math. Inst. Hautes Etudes Sci. ´ 51 (1980), 137-173.
10. A. N. Kolmogorov, New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces, Doklady of Russian Academy of Sciences, 119, N5 (1958),861-864.
11. C. Luo, Y. Zhao, The Brin-Katok formula for dynamical systems admitting mistakes,Dynamical Systems 36 (2021), 560-571.
12. Y. Pesin, Dimension theory in dynamical systems: Contemporary views and applications,The University of Chicago Press, ChicagoIL, 1977.
13. C. Pfister, W. Sullivan, On the topological entropy of saturated sets, Ergod. Theory Dynam. Syst. 27 (2007), 929-956.
14. M. Rahimi, A. Assari, On local metric pressure of dynamical systems, Period Math Hung 82, 223-230 (2021).
15. M. Rahimi and A. Assari, A note on entropy of finitely ergodic compact topological dynamical systems, J. Finsler Geom. Appl. 2(1) (2021), 142-150.
16. M. Rahimi and A. Assari, A family of graphs generated by Hilbert space, J. Finsler Geom. Appl. 6(1) (2025), 114-124.
17. M. Rahimi, M. R. Mozaffari and A. Tayebi, A Mathematical Modeling of n-State Systems,Physica A: Statistical Mechanics and its Applications, 674 (2025) 130727.
18. D. Ruelle, Statistical mechanics on compact set with Zν action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251.
19. D. Ruelle, Thermodynamic formalism, Cambridge Mathematical Library. Cambridge
University Press, second edition, 2004. The mathematical structures of equilibrium statistical mechanics.
20. Ya. G. Sinai, On the notion of entropy of a dynamical system. Doklady of Russian Academy of Sciences, 124(1959), 768-771.
21. D. Thompson, Irregular sets, the beta-transformation and the almost specification property, Trans. Amer. Math. Soc. 364(2012), 5395-5414.
22. P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97(1975), 937-971.
23. D. Zheng, E. Chen, and J. Yang, On large deviations for amenable group actions, Discrete Contin. Dyn. Syst. Ser. A 36(12) (2017), 7191-7206.
24. Y. J. Zhu, Two Notes on Measure-Theoretic Entropy of Random Dynamical Systems,Acta Math. Sin. (Engl. Ser.), 25(6) (2009), 961-970. 
Journal of Finsler Geometry and its Applications

Articles in Press, Accepted Manuscript
Available Online from 10 June 2026

Files

Share

How to cite

Statistics