Distribution of Periods of Closed Trajectories in Exponentially Shrinking Intervals

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 ${(x - e^{-delta x}, x + e^{-delta x}), ; delta > 0, ; x to + infty.}${(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.