Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows

    Research output: Contribution to journalArticle

    30 Citations (Scopus)

    Abstract

    The paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional (visibility) condition. Using a modification of the technique of Dolgopyat, we get spectral estimates for the Ruelle operator related to a Markov family for the nonwandering (trapping) set of the flow similar to those of Dolgopyat in the case of transitive Anosov flows on compact manifolds with smooth jointly nonintegrable horocycle foliations. As a consequence, we get exponential decay of correlation for Holder continuous potentials on the nonwandering set. Combining the spectral estimate for the Ruelle operator with an argument of Pollicott and Sharp, we also derive the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a half-plane Re (s) <h(T) - epsilon, where hT is the topological entropy of the billiard flow, and an asymptotic formula with an error term for the number pi(lambda) of closed orbits of least period lambda > 0.
    Original languageEnglish
    Pages (from-to)715-759
    JournalAmerican Journal of Mathematics
    Volume123
    DOIs
    Publication statusPublished - 2001

    Fingerprint Dive into the research topics of 'Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows'. Together they form a unique fingerprint.

    Cite this