Almost sure convergence of maxima for chaotic dynamical systems
Holland, Mark P.; Nicol, Matthew; Török, János
Date: 21 March 2016
Publisher
Cornell University Library
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Abstract
Suppose (f,X,ν) is a measure preserving dynamical system and ϕ:X→R is an observable with some degree of regularity. We investigate the maximum process M n :=max{X 1 ,…,X n } , where X i =ϕ∘f i is a time series of observations on the system. When M n →∞ almost surely, we establish results on the almost sure growth rate, namely the ...
Suppose (f,X,ν) is a measure preserving dynamical system and ϕ:X→R is an observable with some degree of regularity. We investigate the maximum process M n :=max{X 1 ,…,X n } , where X i =ϕ∘f i is a time series of observations on the system. When M n →∞ almost surely, we establish results on the almost sure growth rate, namely the existence (or otherwise) of a sequence u n →∞ such that M n /u n →1 almost surely. The observables we consider will be functions of the distance to a distinguished point x ~ ∈X . Our results are based on the interplay between shrinking target problem estimates at x ~ and the form of the observable (in particular polynomial or logarithmic) near 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.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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