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

We consider a conservative ergodic measure-preserving transformation $T$ of a $\sigma$-finite measure space $(X,\mathcal{B},\mu )$ with $\mu (X)=\mathrm{\infty }$. Given an observable $f:X\to \mathbb{R}$ we study the almost sure asymptotic behaviour of the Birkhoff sums $S_{N}f(x):={\sum }_{j=1}^N(f\circ T^{j-1})(x)$. In infinite ergodic theory it is well known that the asymptotic behaviour of $S_{N}f(x)$ strongly depends on the point $x\in X$, and if $f\in L^1(X,\mu )$, then there exists no real valued sequence $(b(N))$ such that $\lim_{N\to\infty} S_Nf(x)/b(N)=1$ 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 $(\alpha (N))$ and $m\colon X\times $\mathbb{N}$\to$\mathbb{N}$$suchthatfor$f\in L^1(X,\mu)$wehave$\lim_{N\to\infty} S_{N+m(x,N)}f(x)/\alpha(N)=1$for$\mu$-a\mathrm{.}e\mathrm{.}$x\in X$. Moreover if $f\in ̸L^1(X,\mu )$ we give conditions on the induced observable such that there exists a sequence $(G(N))$ depending on $f$, for which ${\lim }_{N\to \mathrm{\infty }}{S}_{N}f(x)\mathrm{/}G(N)=1$ holds for $\mu$-a.e. $x\in X$.