English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We establish quantitative results for the statistical be\-ha\-vi\-our of\emph{infinite systems}. We consider two kinds of infinite system: i) aconservative dynamical system $(f,X,\mu)$ preserving a $\sigma$-finite measure$\mu$ such that $\mu(X)=\infty$; ii) the case where $\mu$ is a probabilitymeasure but we consider the statistical behaviour of an observable $\phi\colonX\to[0,\infty)$ which is non-integrable: $\int \phi \, d\mu=\infty$.  In the first part of this work we study the behaviour of Birkhoff sums ofsystems of the kind ii). For certain weakly chaotic systems, we show that thesesums can be strongly oscillating. However, if the system has superpolynomialdecay of correlations or has a Markov structure, then we show this oscillationcannot happen. In this case we prove asymptotic relations between the behaviourof $\phi $, the local dimension of $\mu$, and on the growth of Birkhoff sums(as time tends to infinity). We then establish several important consequenceswhich apply to infinite systems of the kind i). This includes showing anomalousscalings in extreme event limit laws, or entrance time statistics. We apply ourfindings to non-uniformly hyperbolic systems preserving an infinite measure,establishing anomalous scalings in the case of logarithm laws of entrancetimes, dynamical Borel--Cantelli lemmas, almost sure growth rates of extremes,and dynamical run length functions.

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