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.

It may be more appropriate to define these notations in the skew-product system $(X\times\mathbb{R},\mu\times m)$, with $(T,\phi):(x,t)\mapsto (Tx,t+\phi(x))$. But above definitions also looks natural :)

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. $x$.

My question is, how to prove these two properties?

Difference with Birkhoff Ergodic Theorem: $\frac{\phi_n(x)}{n}\to0$, but doesn't tell how $\phi_n(x)$ behaves.

Thank you!

Just recalled how to prove the recurrence in (2):

Let $C=\max_{x\in X}|\phi(x)|>0$. Then we divide $X$ into three subsets:

(recurrent part) $x\in X_r$ if $\phi_n(x)\in[-C,C]$ infinitely often;

(positive part) $x\in X_+$ if $\phi_n(x)\ge C$ for all $n$ large;

(negative part) $x\in X_-$ if $\phi_n(x)\le -C$ for all $n$ large.

Moreover both three are invariant. We need to show $X_{\pm}$ are of zero measure. If $\mu(X_+)>0$, then $\mu(X_+)=1$ (by ergodicity). So

$$0=\int\phi d\mu=\int\phi_n d\mu=\lim_{n\to\infty}\int \phi_n d\mu\ge\int\liminf_{n\to\infty}\phi_n d\mu\ge C>0,$$ contradicts in itself and hence $\mu(X_+)=0$. Similarly we have $\mu(X_-)=0$ and hence $\mu(X_r)=1$.

So (2) follows from (1).

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