PETKOV, Vesselin; STOYANOV, Luchezar

doi:10.1007/s00023-020-00956-8

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PETKOV, Vesselin
Institut de Mathématiques de Bordeaux [IMB]

STOYANOV, Luchezar
The University of Western Australia [UWA]

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Annales Henri Poincaré. 2020, vol. 21, n° 12, p. 3791-3834

Springer Verlag

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For hyperbolic flows $\varphi_t$ we examine the Gibbs measure of points $w$ for which $$\int_0^T G(\varphi_t w) dt - a T \in (- e^{-\epsilon n}, e^{- \epsilon n})$$ as $n \to \infty$ and $T \geq n$, provided $\epsilon > 0$ is sufficiently small. This is similar to local central limit theorems. The fact that the interval $(- e^{-\epsilon n}, e^{- \epsilon n})$ is exponentially shrinking as $n \to \infty$ leads to several difficulties. Under some geometric assumptions we establish a sharp large deviation result with leading term $C(a) \epsilon_n e^{\gamma(a) T}$ and rate function $\gamma(a) \leq 0.$ The proof is based on the spectral estimates for the iterations of the Ruelle operators with two complex parameters and on a new Tauberian theorem for sequence of functions $g_n(t)$ having an asymptotic as $ n \to \infty$ and $t \geq n.$< Leer menos

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hyperbolic flows

large deviations

Tauberian theorem for sequence of functions

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Sharp large deviations for hyperbolic flows

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