Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables

Abstract

<p style='text-indent:20px;'>We establish quantitative results for the statistical behaviour of <i>infinite systems</i>. We consider two kinds of infinite system: <p style='text-indent:20px;'>ⅰ) a conservative dynamical system <inline-formula><tex-math id="M2">\begin{document}$ (f,X,\mu) $\end{document}</tex-math></inline-formula> preserving a <inline-formula><tex-math id="M3">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-finite measure <inline-formula><tex-math id="M4">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M5">\begin{document}$ \mu(X) = \infty $\end{document}</tex-math></inline-formula>; <p style='text-indent:20px;'>ⅱ) the case where <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is a probability measure but we consider the statistical behaviour of an observable <inline-formula><tex-math id="M8">\begin{document}$ \phi\colon X\to[0,\infty) $\end{document}</tex-math></inline-formula> which is non-integrable: <inline-formula><tex-math id="M9">\begin{document}$ \int \phi \, d\mu = \infty $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove a general relation between the behaviour of <inline-formula><tex-math id="M10">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>, the local dimension of <inline-formula><tex-math id="M11">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>, and the scaling rate of the growth of Birkhoff sums of <inline-formula><tex-math id="M12">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> as time tends to infinity. We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings for the power law behaviour of entrance times (also known as logarithm laws), dynamical Borel–Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.