Title: On effective Birkhoff's ergodic theorem for computable actions of amenable groups,
Abstract: We introduce computable actions of computable groups and prove the following
versions of effective Birkhoff's ergodic theorem. Let $\Gamma$ be a computable
amenable group, then there always exists a canonically computable tempered
two-sided F{\o}lner sequence $(F_n)_{n \geq
1}$ in $\Gamma$. For a computable, measure-preserving, ergodic action of
$\Gamma$ on a Cantor space $\{0,1\}^{\mathbb N}$ endowed with a computable
probability measure $\mu$, it is shown that for every bounded lower
semicomputable function $f$ on $\{0,1\}^{\mathbb N}$ and for every Martin-Löf
random $\omega \in \{0,1\}^{\mathbb N}$ the equality \[ \lim\limits_{n \to
\infty} \frac{1}{|F_n|} \sum\limits_{g \in F_n} f(g \cdot \omega) = \int\limits
f d \mu \] holds, where the averages are taken with respect to a canonically
computable tempered two-sided F{\o}lner sequence $(F_n)_{n \geq
1}$. We also prove the same identity for all lower semicomputable $f$'s in
the special case when $\Gamma$ is a computable group of polynomial growth and
$F_n:=\mathrm{B}(n)$ is the F{\o}lner sequence of balls around the neutral
element of $\Gamma$.