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Suppose $(f,\mathcal{X},\mu)$ is a measure preserving dynamical system and$\phi \colon \mathcal{X} \to \mathbb{R}$ a measurable function. Consider themaximum process $M_n:=\max\{X_1 \ldots,X_n\}$, where $X_i=\phi\circ f^{i-1}$ isa time series of observations on the system. Suppose that $(u_n)$ is anon-decreasing sequence of real numbers, such that $\mu(X_1>u_n)\to 0$. Forcertain dynamical systems, we obtain a zero--one measure dichotomy for$\mu(M_n\leq u_n\,\textrm{i.o.})$ depending on the sequence $u_n$. Specificexamples are piecewise expanding interval maps including the Gauss map. For thebroader class of non-uniformly hyperbolic dynamical systems, we makesignificant improvements on existing literature for characterising thesequences $u_n$. Our results on the permitted sequences $u_n$ are commensuratewith the optimal sequences (and series criteria) obtained by Klass (1985) fori.i.d. processes. Moreover, we also develop new series criteria on thepermitted sequences in the case where the i.i.d. theory breaks down. Ouranalysis has strong connections to specific problems in eventual always hittingtime statistics and extreme value theory.

Dichotomy results for eventually always hitting time statistics and almost sure growth of extremes