Spectral properties of the Ruelle operator for product-type potentials on shift spaces:

L. Cioletti, M. Denker, A. O. Lopes, M. Stadlbauer

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4 Scopus citations

Abstract

We study a class of potentials f on one-sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness property of these quantities is also discussed, and it is shown that there always exists a Bernoulli equilibrium state even if f does not satisfy Bowen's condition. We apply these results to potentials f:{-1,1}N→R of the form f(x1,x2,⋯)=x1+2-γx2+3-γx3++n-γxn+with γ>1. For 3/2<γ≤2, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist almost surely (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.

Original languageEnglish (US)
Pages (from-to)684-704
Number of pages21
JournalJournal of the London Mathematical Society
Volume95
Issue number2
DOIs
StatePublished - Apr 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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