SpringerLink
Log in

On Relationships between the Spectral Potential of Transfer Operators, \(\boldsymbol t\)-Entropy, Entropy and Topological Pressure

  • Research Articles
  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

The paper is devoted to the analysis of relationships between principal objects of the spectral theory of dynamical systems (transfer and weighted shift operators) and basic characteristics of information theory and thermodynamic formalism (entropy and topological pressure). We present explicit formulas linking these objects with the \(t\)-entropy and spectral potential. Herewith we uncover the role of inverse rami-rate, the forward entropy along with an essential set, and the property of noncontractibility of a dynamical system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. B. Antonevich, “Linear Functional Equations. Operator Approach”, Oper. Theory Adv. Appl., 83 (1996).

    MathSciNet  MATH  Google Scholar 

  2. A. B. Antonevich, V. I. Bakhtin, and A. V. Lebedev, “On \(t\)-Entropy and Variational Principle for the Spectral Radii of Transfer and Weighted Shift Operators”, Ergodic Theory Dynam. Systems, 31 (2011), 995–1042.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. B. Antonevich, V. I. Bakhtin, and A. V. Lebedev, “Crossed Product of a \(C^*\)-Algebra by an Endomorphism, Coefficient Algebras and Transfer Operators”, Sb. Math., 202:9 (2011), 1253–1283.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. B. Antonevich, V. I. Bakhtin, and A. V. Lebedev, “A Road to the Spectral Radius of Transfer Operators”, Contemp. Math., 567 (2012), 17–51.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. B. Antonevich, V. I. Bakhtin, A. V. Lebedev, and D. S. Sarzhinsky, “Legendre Analysis, Thermodynamic Formalizm and Spectra of Perron–Frobenius Operators”, Dokl. Math., 67 (2003), 343–345.

    MATH  Google Scholar 

  6. A. Antonevich, M. Belousov, and A. Lebedev, Functional Differential Equations: II. C*-Applications: Part 1: Equations with Continuous Coefficients, Pitman Monographs and Surveys in Pure and Applied Mathematics, 94, Addison-Wesley, Longman, Harlow, England, 1998.

    MATH  Google Scholar 

  7. A. Antonevich, M. Belousov, and A. Lebedev, Functional Differential Equations: II. C*-Applications: Part 2: Equations with Discontinuous Coefficients and Boundary Value Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 95, Addison-Wesley, Longman, Harlow, England, 1998.

    MATH  Google Scholar 

  8. A. Antonevich and A. Lebedev, Functional Differential Equations: I. \(C^*\)-Theory, vol. 70, Longman Scientific & Technical, Pitman Monographs and Surveys in Pure and Applied Mathematics, 1994.

    MATH  Google Scholar 

  9. V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific, River Edge, NJ, 2000.

    Book  MATH  Google Scholar 

  10. V. I. Bakhtin, “On t-Entropy and Variational Principle for the Spectral Radius of Weighted Shift Operators”, Ergodic Theory Dynam. Systems, 30 (2010), 1331–1342.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Bardadin and B. K. Kwaśniewski, “Spectrum of Weighted Isometries: \(C^*\)-Algebras, Transfer Operators and Topological Pressure”, Israel J. Math., 246:1 (2021), 149–210.

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Bardadyn, B. K. Kwaśniewski, K. S. Kurnosenko, and A. V. Lebedev, “\(t\)-Entropy Formulae for Concrete Classes of Transfer Operators”, J. Belarusian State University, Mathematics and Informatics, :3 (2019), 122–128.

    Article  MATH  Google Scholar 

  13. V. I. Bakhtin and A. V. Lebedev, “A New Definition of \(t\)-Entropy for Transfer Operators”, Entropy, 573:19 (2017), 1–6.

    MathSciNet  Google Scholar 

  14. V. I. Bakhtin and A. V. Lebedev, “Entropy Statistic Theorem and Variational Principle for t-Entropy Are Equivalent”, J. Math. Anal. Appl., 474 (2019), 59–71.

    Article  MathSciNet  MATH  Google Scholar 

  15. V. I. Bakhtin and A. V. Lebedev, “Sup-Sums Principles for F-Divergence and a New Definition for t-Entropy”, J. Theoret. Probab., (2020).

    Google Scholar 

  16. R. Bowen, “Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms”, Lect. Notes in Math., 470 (1975).

    ADS  MathSciNet  MATH  Google Scholar 

  17. V. Didenko, “Estimates of the Spectral Radius of Refinement and Subdivision Operators with Isotropic Dilations”, J. Operator Theory, 58 (2007), 3–22.

    MathSciNet  MATH  Google Scholar 

  18. E. I. Dinaburg, “A Correlation between Topological Entropy and Metric Entropy”, Dokl. Akad. Nauk SSSR, 190:1 (1970), 19–22.

    MathSciNet  Google Scholar 

  19. R. Exel, “A New Look at the Crossed-Product of a \(C^*\)-Algebra by an Endomorphism”, Ergodic Theory Dynam. Systems, 23:6 (2003), 1733–1750.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. H. Fan and Y. P. Jiang, “On Ruelle–Perron–Frobenius Operators. I. Ruelle’s Theorem”, Comm. Math. Phys., 223 (2001), 125–141.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. N. Tim and T. Goodman, “Relating Topological Entropy and Measure Entropy”, Bull. London. Math. Soc., 3 (1971), 176–180.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Kitaev, “Fredholm Determinants for Hyperbolic Diffeomorphisms of Finite Smoothness”, Nonlinearity, 12 (1999), 141–179.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. A. K. Kitover, “Spectrum of Automorphisms with Weight and the Kamowitz–Scheinberg Theorem”, Funct. Anal. Appl., 13:1 (1979), 57–58.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Knauf and Ya. Sinai, Classical Nonintegrability, Quantum Chaos, Birkhauser, Basel, 1997.

    Book  MATH  Google Scholar 

  25. B. K. Kwaśniewski, “On Transfer Operators for \(C^*\)-Dynamical Systems”, Rocky Mountain J. Math., 42:3 (2012), 919–938.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. K. Kwaśniewski and A. V. Lebedev, “Crossed Products by Endomorphisms and Reduction of Relations in Relative Cuntz-Pimsner Algebras”, J. Funct. Anal., 264:8 (2013), 1806–1847.

    Article  MathSciNet  MATH  Google Scholar 

  27. B. K. Kwaśniewski and A. V. Lebedev, “Variational Principles for Spectral Radius of Weighted Endomorphisms of \(C(X,D)\)”, Trans. Amer. Math. Soc., 373:4 (2020), 2659–2698.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. V. Lebedev, “On the Invertiblity of Elements in \(C^*\)-Algebras Generated by Dynamical Systems”, Russian Math. Surveys, 34:4 (1979), 174–175.

    Article  ADS  MATH  Google Scholar 

  29. A. Lebedev and O. Maslak, The Spectral Radius of a Weighted Shift Operator, Variational Principles, Entropy and Topological Pressure, Spectral and Evolutionary Problems. Proceedings of the Eighth Crimean Autumn Mathematical School Symposium (Simferopol, 1998), Tavria Publishers, Moscow,.

    Google Scholar 

  30. Ju. D. Latushkin and A. M. Stepin, “Weighted Shift Operators on a Topological Markov Chain”, Funct. Anal. Appl., 22:4 (1988), 330–331.

    Article  MathSciNet  MATH  Google Scholar 

  31. W. Parry and M. Pollicott, “Zeta Functions and Closed Orbits for Hyperbolic Systems”, Asterisque, :187–188 (1990), 1–268.

    Google Scholar 

  32. F. Przytycki and M. Urbanski, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series 371, Cambridge University Press, 2010.

    Book  MATH  Google Scholar 

  33. H. Rough, “Generalized Fredholm Determinants and Selberg Zeta Functions for Axiom A Dynamical Systems”, Ergodic Theory Dynam. Systems, 16 (1996), 805–819.

    Article  MathSciNet  Google Scholar 

  34. D. Ruelle, “Statistical Mechanics on a Compact Set with \(Z^\nu\) Action Satisfying Expansiveness and Specification”, Trans. Amer. Math. Soc., 185 (1973), 237–251.

    Article  MathSciNet  MATH  Google Scholar 

  35. D. Ruelle, Thermodynamic Formalism. Encyclopedia of Math. and its Appl., vol. 5, Reading, Mass: Addison-Wesley, 1978.

    Google Scholar 

  36. D. Ruelle, “The Thermodynamic Formalism for Expanding Maps”, Comm. Math. Phys., 125 (1989), 239–262.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. D. Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (CRM Monograph Series, 4), American Mathematical Society, Providence, RI, 1991.

    Google Scholar 

  38. P. Walters, “A Variational Principle for the Pressure on Continuous Transformations”, Amer. J. Math., 97:4 (1975), 937–971.

    Article  MathSciNet  MATH  Google Scholar 

  39. P. Walters, “Invariant Measures and Equilibrium States for Some Map**s which Expand Distances”, Trans. Amer. Math. Soc., 236 (1978), 121–153.

    Article  MathSciNet  MATH  Google Scholar 

  40. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York—Berlin, 1982.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Belarusian State University, Minsk, Belarus

    V. I. Bakhtin & A. V. Lebedev

Authors
  1. V. I. Bakhtin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Lebedev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Bakhtin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bakhtin, V.I., Lebedev, A.V. On Relationships between the Spectral Potential of Transfer Operators, \(\boldsymbol t\)-Entropy, Entropy and Topological Pressure. Russ. J. Math. Phys. 30, 1–24 (2023). https://doi.org/10.1134/S1061920823010016

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920823010016

Search

Navigation