Let $\varphi:G\times X\to X$ be a continuous action of finitely generated group $G=\langle S \rangle$ on compact metric space $X$. Also, assume that $h:X\to X$ be a homeomorphism with $d(h(x), x)<\delta$ for all $x\in X$.
Assume that $\psi:G\times X\to X$ is generated by $h\circ \varphi_s$ for all $s\in S$.
Q1. What can say about the size $d(\psi(s, x), \varphi(s, x))$ for all $x\in X$?
Q2. Is it true $\psi(s, x)= h\circ \varphi_s(x)$? If it is true, what can say about equality $\psi(s_0s_1, x)= \psi(s_0, \psi(s_1, x))$?
Note that $\psi(s_0, \psi(s_1, x))= h\circ\varphi_{s_0}\circ h\circ\varphi_{s_1}(x))$