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

Stefano Galatolo; Mark Holland; Tomas Persson; Yiwei Zhang

doi:10.3934/dcds.2020341

Article Contents

2021, Volume 41, Issue 4: 1799-1841. Doi: 10.3934/dcds.2020341

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

1.
Dipartimento di Matematica, Via Buonattoti 1, 56127 Pisa, Italy
2.
Mathematics (CEMPS), Harrison Building (327), North Park Road, EXETER, EX4 4QF, United Kingdom
3.
Centre for Mathematical Sciences, Lund University, Box 118,221 00 Lund, Sweden
4.
School of Mathematics and Statistics, Center for Mathematical Sciences, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Sciences and Technology, Wuhan 430074, China

Received: March 2020

Revised: August 2020

Early access: October 2020

Published: April 2021

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Abstract

We establish quantitative results for the statistical behaviour of infinite systems. We consider two kinds of infinite system:

ⅰ) a conservative dynamical system $ (f,X,\mu) $ preserving a $ \sigma $-finite measure $ \mu $ such that $ \mu(X) = \infty $;

ⅱ) the case where $ \mu $ is a probability measure but we consider the statistical behaviour of an observable $ \phi\colon X\to[0,\infty) $ which is non-integrable: $ \int \phi \, d\mu = \infty $.

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 $ \phi $, the local dimension of $ \mu $, and the scaling rate of the growth of Birkhoff sums of $ \phi $ 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.

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Mathematics Subject Classification: Primary: 37A40, 37A50, 60G70; Secondary: 37A25, 37D25.

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Figure 1. An intermittent map, with induced map in top right quadrant

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