Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

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:X\to \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\, (f\circ 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) $\end{document}</tex-math></inline-formula> strongly depends on the point <inline-formula><tex-math id="M8">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>, and if <inline-formula><tex-math id="M9">\begin{document}$ f\in 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_{N\to\infty} 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)) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ m\colon X\times \mathbb N\to \mathbb N $\end{document}</tex-math></inline-formula> such that for <inline-formula><tex-math id="M14">\begin{document}$ f\in L^1(X,\mu) $\end{document}</tex-math></inline-formula> we have <inline-formula><tex-math id="M15">\begin{document}$ \lim_{N\to\infty} 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 $\end{document}</tex-math></inline-formula>-a.e. <inline-formula><tex-math id="M17">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>. Instead in the case <inline-formula><tex-math id="M18">\begin{document}$ f\not\in 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_{N\to\infty} S_{N}f(x)/G(N) = 1 $\end{document}</tex-math></inline-formula> holds for <inline-formula><tex-math id="M22">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>-a.e. <inline-formula><tex-math id="M23">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>.</p>