Let $G=\langle S \rangle$ be a finitely generated (not necessarily abelian) group and $\varphi:G\times X\to X$ be a action such that $\varphi_g:X\to X$ is homeomorphism. Let $g=s_r\ldots s_1$, $s_i\in S$, be given. For every $\epsilon>0$ there is $\delta>0$ such that for every $t=s_j\ldots s_1$ or $s_j^{-1}\ldots s_1^{-1}$, for some $1\leq j\leq r$ we have \begin{equation} d(a, b)<\delta \Rightarrow d(\varphi_t(a), \varphi_t(b))<\epsilon \end{equation} Consider $\{x_n\}_{n\in \mathbb{Z}}\subseteq X$ with $d(\varphi(g, x_n), x_{n+1})<\delta$. Define $\{z_n\}_{n\in \mathbb{Z}}$ by $z_{rk}=x_k$ for all $r\in\mathbb{Z}$ and $z_{rk+j+1}=\varphi(s_{j+1}, z_{rk+j})$ for all $0\leq j< r-1$.
For every $t\in G$ find an element $v\in G$ of minimal length such that $t = vw$ where $w=s_j\ldots s_1(g)^k$ for some $1\leq j\leq r$ and $k\geq 0$. Or $w=s_{r-j+1}^{-1}\ldots s_r^{-1}$ for some $1\leq j\leq r$ and $k\geq 0$.
If $w=s_j\ldots s_1(g)^k$, then define $y_t=\varphi(v, z_{rk+j})$ and if $w=s_{r-j+1}^{-1}\ldots s_r^{-1}$, then define $y_t= \varphi(v, z_{-rk-j})$.
In a paper, author claim that $d(y_{st}, \varphi(s, y_t))<\epsilon$ for all $t\in G$ and all $s\in S$. But it is not clear for me. Please help me to know it.