Almost sure convergence of maxima for chaotic dynamical systems

Suppose (f,X,ν)(f,X,ν) is a measure preserving dynamical system and ϕ:X→Rϕ:X→R is an observable with some degree of regularity. We investigate the maximum process Mn:=max(X1,…,Xn)Mn:=max(X1,…,Xn), where Xi=ϕ∘fiXi=ϕ∘fi is a time series of observations on the system. When Mn→∞Mn→∞ almost surely, we establish results on the almost sure growth rate, namely the existence (or otherwise) of a sequence un→∞un→∞ such that Mn/un→1Mn/un→1 almost surely. For a wide class of non-uniformly hyperbolic dynamical systems we determine where such an almost sure limit exists and give examples where it does not.