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Journal of Dynamical and Control Systems

, Volume 19, Issue 3, pp 327–347 | Cite as

On asymptotic properties of equilibrium measures corresponding to finite submatrices of infinite nonnegative matrices

  • B. M. Gurevich
  • O. R. Novokreschenova
Article
  • 85 Downloads

Abstract

There is a canonical way to assign a translation invariant Markov measure to every finite irreducible nonnegative matrix. This measure is defined on a sequence space, translation invariant, and satisfies a variational principle (due to the last property it is said to be equilibrium). The same can be done for some infinite nonnegative matrices. If A is such a matrix, one can consider an increasing sequence of its finite irreducible submatrices A n that tends to A in a natural sense, and ask if the equilibrium measure μ An assigned to A n converges to the equilibrium measure μ A assigned to A. For two classes of matrices A, the answer does not depend on {A n }: for one of these classes, μ An → μ A , for the other one, μ An converges to the zero measure. We describe, in geometric terms, a third class located ‘between’ the above two for which the situation is also intermediate: for some sequences {A n } the asymptotic behavior of μ An is as in the first class, while for some other sequences the behavior is as in the second one.

Key words and phrases

Nonnegative matrix loaded graph equilibrium measure 

2000 Mathematics Subject Classification

37A60 37D35 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Institute for Information Transmission ProblemsMoscowRussia

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