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

Published online by Cambridge University Press:  25 June 2010

A. B. ANTONEVICH ,
V. I. BAKHTIN  and
A. V. LEBEDEV
A. B. ANTONEVICH
Affiliation:
University of Bialystok, Bialystok, Poland & Belarus State University, Minsk, Belarus (email: [email protected], [email protected])
V. I. BAKHTIN
Affiliation:
Belarus State University, Minsk, Belarus (email: [email protected])
A. V. LEBEDEV
Affiliation:
University of Bialystok, Bialystok, Poland & Belarus State University, Minsk, Belarus (email: [email protected], [email protected])
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Abstract

The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or a topological Markov chain. As the main summands, these principles contain the integrals over invariant measures and the Kolmogorov–Sinai entropy. In the paper we derive the variational principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov–Sinai entropy but a new invariant of entropy type—the t-entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 995 - 1042
Copyright
Copyright © Cambridge University Press 2010

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