The structure of equicontinuous maps

Let $(X,d)$ be a metric space, and $f:X\rightarrow X$ be a continuous map. In this paper we prove that if $R(f)$ is compact, and $\omega (x,f)\not =\emptyset$ for all $x\in X$, then $f$ is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism $h$ and a non-expanding map $g$ that is pointwise convergent to a fixed point $v_{0}$ such that $f$ is uniformly conjugate to a subsystem $(h\times g)\vert S$ of the product map $h\times g$. In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.