Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T00:29:30.731Z Has data issue: false hasContentIssue false
  • English
  • Français

On t-entropy and variational principle for the spectral radius of weighted shift operators

Published online by Cambridge University Press:  24 November 2009

V. I. BAKHTIN
V. I. BAKHTIN*
Affiliation:
Department of Mechanics and Mathematics, Belarus State University, Nezavisimosty av. 4, Minsk 220050, Belarus (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we introduce a new functional invariant of discrete time dynamical systems—the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on L1(X,m) generated by the dynamical system. This result is called the variational principle and is similar to the classical variational principle for the topological pressure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1331 - 1342
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Antonevich, A. B., Bakhtin, V. I. and Lebedev, A. V.. Variational principle for the spectral radius of weighted shift operators in Lebesgue spaces. Tr. Inst. Mat., Minsk 5 (2000), 13–17 (in Russian).Google Scholar
[2]Antonevich, A. B., Bakhtin, V. I. and Lebedev, A. V.. A variational principle for the spectral radius of weighted shift operators and weighted mathematical expectation operators. Dokl. Nats. Akad. Nauk Belarusi 44(6) (2000), 7–10 (in Russian).Google Scholar
[3]Antonevich, A., Bakhtin, V. and Lebedev, A.. Thermodynamics and spectral radius. Nonlinear Phenom. Complex Syst. 4(4) (2001), 318–321.Google Scholar
[4]Antonevich, A. B., Bakhtin, V. I. and Lebedev, A. V.. T-entropy and variational principle for the spectral radius of transfer and weighted shift operators. Preprint, 2008, arXiv:0809.3116v2 [math.DS].Google Scholar
[5]Bakhtin, V. I.. T-entropy and variational principle for the spectral radius of weighted shift operators. Preprint, 2008, arXiv:0809.3106v2 [math.DS].Google Scholar
[6]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar