Abstract
For hyperbolic flows over basic sets we study the asymptotic of the number of closed trajectories γ with periods T γ lying in exponentially shrinking intervals \({(x - e^{-\delta x}, x + e^{-\delta x}), \; \delta > 0, \; x \to + \infty.}\) A general result is established which concerns hyperbolic flows admitting symbolic models whose corresponding Ruelle transfer operators satisfy some spectral estimates. This result applies to a variety of hyperbolic flows on basic sets, in particular to geodesic flows on manifolds of constant negative curvature and to open billiard flows.
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Communicated by S. Zelditch
The first author was partially supported by the ANR project NONAA.
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Petkov, V., Stoyanov, L. Distribution of Periods of Closed Trajectories in Exponentially Shrinking Intervals. Commun. Math. Phys. 310, 675–704 (2012). https://doi.org/10.1007/s00220-012-1419-x
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DOI: https://doi.org/10.1007/s00220-012-1419-x