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- W2217635909 abstract "Suppose $(f,mathcal{X},nu)$ is a measure preserving dynamical system and $phi:mathcal{X}tomathbb{R}$ is an observable with some degree of regularity. We investigate the maximum process $M_n:=max{X_1,ldots,X_n}$, where $X_i=phicirc f^i$ is a time series of observations on the system. When $M_ntoinfty$ almost surely, we establish results on the almost sure growth rate, namely the existence (or otherwise) of a sequence $u_ntoinfty$ such that $M_n/u_nto 1$ almost surely. The observables we consider will be functions of the distance to a distinguished point $tilde{x}in mathcal{X}$. Our results are based on the interplay between shrinking target problem estimates at $tilde{x}$ and the form of the observable (in particular polynomial or logarithmic) near $tilde{x}$. We determine where such an almost sure limit exists and give examples where it does not. Our results apply to a wide class of non-uniformly hyperbolic dynamical systems, under mild assumptions on the rate of mixing, and on regularity of the invariant measure." @default.
- W2217635909 created "2016-06-24" @default.
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- W2217635909 date "2015-10-15" @default.
- W2217635909 modified "2023-09-26" @default.
- W2217635909 title "Almost sure convergence of maxima for chaotic dynamical systems" @default.
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- W2217635909 doi "https://doi.org/10.48550/arxiv.1510.04681" @default.
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