Let $G$ be finitely generated group (i.e $G= <S>$  $S=\{ s_1, ...,
s_n\}$) and  $\varphi:G\times M\longrightarrow M$ is an action then
$f:G\longrightarrow M$ is called $\delta$- pseudo orbit if $d(f(sg),
\varphi(s,f(g)))<\delta$ for every $s\in S$, $g\in G$. How can we
define asymptotic pseudo orbit for action $\varphi:G\times
M\longrightarrow M$?