  The structure of the $\omega$-limit sets is thoroughly investigated for the
skew-product semiflow which is generated by a scalar reaction-diffusion
equation \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 uniformly almost
periodic in $t$ and satisfies $f(t,u,u_x)=f(t,u,-u_x)$. We show that any
$\omega$-limit set $\Omega$ contains at most two minimal sets. Moreover, any
hyperbolic $\omega$-limit set $\Omega$ is a spatially-homogeneous $1$-cover of
hull $H(f)$. When $\dim V^c(\Omega)=1$ ($V^c(\Omega)$ is the center space
associated with $\Omega$), it is proved that either $\Omega$ is a
spatially-homogeneous, or $\Omega$ is a spatially-inhomogeneous $1$-cover of
$H(f)$.
