Communications in Mathematical Physics

, Volume 292, Issue 3, pp 637–666 | Cite as

Spectral Gap and Transience for Ruelle Operators on Countable Markov Shifts

  • Van Cyr
  • Omri Sarig


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.


Spectral Radius Equilibrium Measure Invariant Probability Measure Thermodynamic Formalism Positive Continuous Function 
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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Mathematics DepartmentThe Pennsylvania State UniversityUniversity ParkUSA

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