Let $F_2=\langle a, b \rangle$ be a free group and let $f, g:X\to X$ be two homeomorphism on compact metric space with $f\circ g\neq id_X$.
Assume that $\varphi:F_2\times X\to X$ is defined by $\varphi(a, x)= f(x)$ and $\varphi(b, x)= g(x)$. Hence, if $t=a^n b^m$, then $\varphi(t, x)= f^n(g^m(x))$, where $f^n(x)= f\circ\ldots \circ f(x)$, $n$-times.
Also, denote by $\psi$ group action generated by $f, g$.
What is different between $\varphi$ and $\psi$?
Can I defined $\psi$ as a group action of $F_2\times X\to X$?