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

<p style='text-indent:20px;'>We consider a conservative ergodic measure-preserving transformation <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> of id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-finite measure space id="M3">\begin{document}$ (X, {\mathcal B},\mu) with id="M4">\begin{document}$ \mu(X) = \infty $\end{document}</tex-math></inline-formula>. Given an observable id="M5">\begin{document}$ f:X\to \mathbb R $\end{document}</tex-math></inline-formula>, we study the almost sure asymptotic behaviour Birkhoff sums id="M6">\begin{document}$ S_Nf(x) : \sum_{j 1}^N\, (f\circ T^{j-1})(x) In infinite theory it is well known that id="M7">\begin{document}$ strongly depends on point id="M8">\begin{document}$ x\in X and if id="M9">\begin{document}$ f\in L^1(X,\mu) then there exists no real valued sequence id="M10">\begin{document}$ (b(N)) such id="M11">\begin{document}$ \lim_{N\to\infty} S_Nf(x)/b(N) 1 surely. this paper show for dynamical systems strong mixing assumptions induced map finite set, id="M12">\begin{document}$ (\alpha(N)) id="M13">\begin{document}$ m\colon X\times N\to N id="M14">\begin{document}$ have id="M15">\begin{document}$ S_{N+m(x,N)}f(x)/\alpha(N) id="M16">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>-a.e. id="M17">\begin{document}$ Instead in case id="M18">\begin{document}$ f\not\in give conditions id="M19">\begin{document}$ (G(N)) depending id="M20">\begin{document}$ f which id="M21">\begin{document}$ S_{N}f(x)/G(N) holds id="M22">\begin{document}$ id="M23">\begin{document}$ $\end{document}</tex-math></inline-formula>.</p>