Let $(X,D)$ be a compact metric space and $\{f_n\}_{n\in\mathbb{N}}$ be a sequence of homeomorphisms of $(X,d)$. It is easy to see that if $\{f_n\}$ is uniformly convergent then $\{g_n\}$ defined by $g_n=f_{2n}\circ f_{2n-1}$ is also uniformly convergent. Thus, for every $\epsilon>0$ there is $\delta>0$ such that if $d(x,y)<\delta$ then $d(g_n(x), g_n(y))<\epsilon$. This means that uniform convergence of $\{f_n\}$ implies that the family $\{f_{2n}\circ f_{2n-1}\}$ is equicontinuous.

Are there other conditions which imply that the sequence $\{f_{2n}\circ f_{2n-1}\}$ is equicontinuous?