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• W3152936637 abstract "<p style='text-indent:20px;'>We consider a conservative ergodic measure-preserving transformation <inline-formula><tex-math id=M1>begin{document}$ T $end{document}</tex-math></inline-formula> of a <inline-formula><tex-math id=M2>begin{document}$ sigma $end{document}</tex-math></inline-formula>-finite measure space <inline-formula><tex-math id=M3>begin{document}$ (X, {mathcal B},mu) $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=M4>begin{document}$ mu(X) = infty $end{document}</tex-math></inline-formula>. Given an observable <inline-formula><tex-math id=M5>begin{document}$ f:Xto mathbb R $end{document}</tex-math></inline-formula>, we study the almost sure asymptotic behaviour of the Birkhoff sums <inline-formula><tex-math id=M6>begin{document}$ S_Nf(x) : = sum_{j = 1}^N, (fcirc T^{j-1})(x) $end{document}</tex-math></inline-formula>. In infinite ergodic theory it is well known that the asymptotic behaviour of <inline-formula><tex-math id=M7>begin{document}$ S_Nf(x) $end{document}</tex-math></inline-formula> strongly depends on the point <inline-formula><tex-math id=M8>begin{document}$ xin X $end{document}</tex-math></inline-formula>, and if <inline-formula><tex-math id=M9>begin{document}$ fin L^1(X,mu) $end{document}</tex-math></inline-formula>, then there exists no real valued sequence <inline-formula><tex-math id=M10>begin{document}$ (b(N)) $end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id=M11>begin{document}$ lim_{Ntoinfty} S_Nf(x)/b(N) = 1 $end{document}</tex-math></inline-formula> almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence <inline-formula><tex-math id=M12>begin{document}$ (alpha(N)) $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=M13>begin{document}$ mcolon Xtimes mathbb Nto mathbb N $end{document}</tex-math></inline-formula> such that for <inline-formula><tex-math id=M14>begin{document}$ fin L^1(X,mu) $end{document}</tex-math></inline-formula> we have <inline-formula><tex-math id=M15>begin{document}$ lim_{Ntoinfty} S_{N+m(x,N)}f(x)/alpha(N) = 1 $end{document}</tex-math></inline-formula> for <inline-formula><tex-math id=M16>begin{document}$ mu $end{document}</tex-math></inline-formula>-a.e. <inline-formula><tex-math id=M17>begin{document}$ xin X $end{document}</tex-math></inline-formula>. Instead in the case <inline-formula><tex-math id=M18>begin{document}$ fnotin L^1(X,mu) $end{document}</tex-math></inline-formula> we give conditions on the induced observable such that there exists a sequence <inline-formula><tex-math id=M19>begin{document}$ (G(N)) $end{document}</tex-math></inline-formula> depending on <inline-formula><tex-math id=M20>begin{document}$ f $end{document}</tex-math></inline-formula>, for which <inline-formula><tex-math id=M21>begin{document}$ lim_{Ntoinfty} S_{N}f(x)/G(N) = 1 $end{document}</tex-math></inline-formula> holds for <inline-formula><tex-math id=M22>begin{document}$ mu $end{document}</tex-math></inline-formula>-a.e. <inline-formula><tex-math id=M23>begin{document}$ xin X $end{document}</tex-math></inline-formula>.</p>" @default.
• W3152936637 created "2021-04-26" @default.
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• W3152936637 date "2022-01-01" @default.
• W3152936637 modified "2023-09-26" @default.
• W3152936637 title "Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems" @default.
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• W3152936637 doi "https://doi.org/10.3934/dcds.2022113" @default.
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