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

Abstract

Suppose (f, X , µ) is a measure preserving dynamical system and φ : X → Ê a measurable function. Consider the maximum process Mn := max{X1, . . . , Xn}, where Xi = φ • f i−1 is a time series of observations on the system. Suppose that (un) is a non-decreasing sequence of real numbers, such that µ(X1 > un) → 0. For certain dynamical systems, we obtain a zero-one measure dichotomy for µ(Mn ≤ un i.o.) depending on the sequence un. Specific examples are piecewise expanding interval maps including the Gauß map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences un. Our results on the permitted sequences un are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory.

Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence

“…The works [94,98,105,111] study the recurrence problem when the lim sup in (4.2) is replaced by lim inf. In particular, [111] proves that for several expanding maps lim inf n→∞ n d (1) n (x, y) ln ln n exists for almost all y.…”

Uniform approximation problems of expanding Markov maps

“…The authors would like to thank Tomas Persson for pointing out that Lebesgue measure statement stated in Example 1 can be concluded from Ref. [15] by a Fubini-based argument. The authors are very grateful to the anonymous referee for his/her valuable comments and suggestions, especially for pointing out that item (2) of Theorem 1.2 is valid in a wider setting and is a direct consequence of [15, Theorem 3.2] and [19, Proposition 1 and Theorem 5].…”