Birkhoff sum and average

Remarks

The same formulas make sense for $f$ with values in $\C$ or in any real vector space. The sums satisfy the cocycle relation $S_{m+n} f = S_m f + (S_n f) \circ T^m$ for all $m, n \ge 1$. If $\mu$ is a $T$-invariant probability measure and $f$ is integrable, then $\int A_n f \, d\mu = \int f \, d\mu$ for every $n$.

Example: let $T(x) = x + 1/2 \bmod 1$ on $[0, 1)$ and $f(x) = x$. For $x \in [0, 1/2)$ the orbit alternates between $x$ and $x + 1/2$, so $S_{2m} f(x) = m(2x + 1/2)$ and $A_{2m} f(x) = x + 1/4$.

Non-example: the space average $\int_0^1 f(x) \, dx = 1/2$ is not a Birkhoff average, since it does not depend on a starting point; in this example $A_{2m} f(x) = x + 1/4$ differs from $1/2$ unless $x = 1/4$.