Second part: if your system is nontrivial, and function $f$ is "smooth", than condition of the form $$\lim_{n\to\infty} \frac 1n\sum_{i=0}^{n-1} f(T^i x) =c$$ for {\bf all} $x$, \ x$, should imply that $$f(x)= g(x)-g(Tx)+c$$ for some function$g$. In this case, one says that$f$is cohomologous to a constant. 1 Second part: if your system is nontrivial, and function$f$is "smooth", than condition of the form $$\lim_{n\to\infty} \frac 1n\sum_{i=0}^{n-1} f(T^i x) =c$$ for {\bf all}$x$, should imply that $$f(x)= g(x)-g(Tx)+c$$ for some function$g$. In this case, one says that$f\$ is cohomologous to a constant.