Thursday 25 Sep 2014AG Dynamics seminar: Rare events and extremal indices via spectral perturbation

Gerhard Keller - University of Erlangen

Harrison 209 16:00-17:00

The first occurence of a rare event $A_\epsilon$ in a time discrete dynamical system can be described in terms of the first hitting time $\tau_\epsilon(x)=inf {i\geq 0 : T^ix \in A_\epsilon}$. In order to justify the adjective "rare" one should assume that $\mu(A_\epsilon)$ is small, for a suitable reference probability measure on $M$. For many dynamical systems it is then known that the law of $\tau_\epsilon$ under $\mu$ is nearly exponential with a parameter $\lambda_\epsilon$ that is closely related to $\mu(A_\epsilon)$. In this talk I will describe how asymptotic expressions for the exponent $\lambda_\epsilon$ of exponential hitting time statistics can be derived from spectral perturbation results for the Perron Frobenius operator of the dynamical system, provided the operator has a spectral gap on a suitable space of (generalized) functions. A simple version of the main formula in the case of piecewise expanding systems is $\lambda_\epsilon=\exp[-\mu(A_\epsilon).\theta.(1+o(1))]$ as $\epsilon\rightarrow 0$ where $\mu$ is the absolutely continuous invariant measure and $\theta$ corresponds to the so called extremal index from time series analysis, a quantity for which I can give a formula in terms of the Perron-
Frobenius operator of the system.

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