  We study the properties of non-wandering points of the following scalar
reaction-diffusion equation on the circle $S^1$, \begin{equation*}
u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2\pi \mathbb{Z},
\end{equation*} where $f$ is independent of $t$ or $T$-periodic in $t$. Assume
that the equation admits a compact global attractor. It is proved that, any
non-wandering point is a limit point of the system (that is, it is a point in
some $\omega$-limit set). More precisely, in the autonomous case, it is proved
that any non-wandering point is either a fixed point or generates a rotating
wave on the circle. In the periodic case, it is proved that any non-wandering
point is a periodic point or generates a rotating wave on a torus.
  In particular, if $f(t,u,-u_x)=f(t,u,u_x)$, then any non-wandering point is a
fixed point in the autonomous case, and is a periodic point in the periodic
case.
