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 $\varphi:G\times M\longrightarrow M$?
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1$\begingroup$ Is $M$ a Riemannian manifold or something of this sort? It might help to add some details, as well as some context for your question. $\endgroup$– Peter CrooksCommented Dec 27, 2014 at 23:46
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$\begingroup$ Is there an existing definition of an asymptotic pseudo-orbit for a $\mathbb{Z}$-action? If so, what is it? $\endgroup$– Ian MorrisCommented Dec 28, 2014 at 13:35
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$\begingroup$ Let $M$ be a compact manifold, In $(M, g)$, $\{x_n\}\subseteq M$ is an asymptotic pseudo orbit if $d(g(x_n), x_{n+1})\longrightarrow 0$ as $|n|\longrightarrow \infty$. $\endgroup$– Ali BarzanouniCommented Dec 28, 2014 at 18:27
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