Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and $\phi\_n(x)=\phi(x)+\cdots+\phi(T^{n-1}x)$ be the induced cocycle.


A point $x\in X$ is said to be $\phi$-transient if $|\phi\_n(x)|\to\infty$ as $n\to\infty$. Otherwise $x$ is said to be $\phi$-recurrent. The following two propositions strengthen the $\phi$-recurrence property:


(1). For $\mu$-a.e. $\phi$-recurrent point $x$, $\displaystyle \liminf\_{n\to\infty}|\phi\_{n}(x)|=0$.


(2). Assume $(T,\mu)$ is ergodic and $\int\phi d\mu=0$. Then $\mu$-a.e. point $x$ is $\phi$-recurrent. In particular $\displaystyle \liminf\_{n\to\infty}|\phi\_{n}(x)|=0$ $\mu$-a.e. $x$.


How to prove these two properties?



Recurrence and transience of cocycle over a dynamical system
I am not quite sure what the question is, but the following result might be a helpful starting point:


Theorem (Giles Atkinson, 1976): Let $T$ be an ergodic invertible measure-preserving transformation of a probability space $(X,\mathcal{F},\mu)$ and let $\phi \colon X \to \mathbb{R}$ be integrable. Then the following statements are equivalent:


1. $\int \phi\,d\mu=0$
2. For all $A \in \mathcal{F}$ such that $\mu(A)>0$, and all $\varepsilon>0$, there exists a nonzero integer $n$ such that $\mu\left(A \cap T^{-n}A \cap \{x \colon |\phi\_n(x)|<\varepsilon\}\right)>0$.


Here we define $\phi\_n(x)=0$ if $n=0$ and $\phi\_n(x)=\phi\_{-n}(T^nx)$ if $n$ is negative. Note that by taking $A=X$ and considering a sequence of values of $\varepsilon$ tending to zero we obtain $\liminf\_{|n|\to \infty} \phi\_n(x)=0$ a.e. 


This falls slightly short of proving the statement (2) in the original post since *a priori* the sequence of integers $n$ along which convergenence to zero takes place might not take infinitely many positive values. I am fairly sure that the sequence can be taken so as to tend to $+\infty$ (indeed it is not unusual to see Atkinson's theorem cited in that form) but Atkinson's original paper unfortunately does not include that statement. I seem to remember that Aaronson's book on infinite ergodic theory includes a proof and so might shed light on this question, but I appear to have mislaid my copy.


(That a.e. $x \in X$ is recurrent in the topological sense (when $X$ is a compact metric space) when $\mu$ is an invariant Borel probability measure is a standard result: see for example Proposition 4.1.18 in Katok and Hasselblatt.)


