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.
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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. |
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