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Spectral Gap and Transience for Ruelle Operators on Countable Markov Shifts

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Abstract

We find a necessary and sufficient condition for the Ruelle operator of a weakly Hölder continuous potential on a topologically mixing countable Markov shift to act with spectral gap on some rich Banach space. We show that the set of potentials satisfying this condition is open and dense for a variety of topologies. We then analyze the complement of this set (in a finer topology) and show that among the three known obstructions to spectral gap (weak positive recurrence, null recurrence, transience), transience is open and dense, and null recurrence and weak positive recurrence have empty interior.

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Authors and Affiliations

  1. Mathematics Department, The Pennsylvania State University, University Park, PA, 16802, USA

    Van Cyr & Omri Sarig

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  1. Van Cyr
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  2. Omri Sarig
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Correspondence to Omri Sarig.

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Communicated by G. Gallavotti

O.S. was partially supported by an NSF grant DMS-0652966 and by an Alfred P. Sloan Research Fellowship.

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Cyr, V., Sarig, O. Spectral Gap and Transience for Ruelle Operators on Countable Markov Shifts. Commun. Math. Phys. 292, 637–666 (2009). https://doi.org/10.1007/s00220-009-0891-4

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  • DOI: https://doi.org/10.1007/s00220-009-0891-4

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