Almost sure convergence of maxima for chaotic dynamical systems

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Suppose (f,X,ν) is a measure preserving dynamical system and ϕ:X is an observable with some degree of regularity. We investigate the maximum process Mₙ:={X₁,,Xₙ}, where Xᵢ=ϕ∘ fⁱ is a time series of observations on the system. When Mₙ→∞ almost surely, we establish results on the almost sure growth rate, namely the existence (or otherwise) of a sequence uₙ→∞ such that Mₙ/uₙ→ 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.