Lifting properties of minimal sets for parabolic equations on 𝑆¹ with reflection symmetry

We consider the skew-product semiflow generated by the following parabolic equation: u t = u x x + f ( t , u , u x ) , t > 0 , x ∈ S 1 = R / 2 π Z , \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 ( t , u , u x ) = f ( t , u , − u x ) f(t,u,u_x)=f(t,u,-u_x) . It is proved that the flow on uniquely ergodic minimal set M M is topologically conjugate to a subflow on R × H ( f ) \mathbb {R}\times H(f) and M M is uniquely ergodic if and only if the set consisting of 1 1 -cover points of H ( f ) H(f) has full measure. It is further proved that any minimal set M M is almost automorphic provided that f f is uniformly almost automorphic. Moreover, for any almost automorphic solution u ( t , x ) u(t,x) contained in M M , the frequency module M ( u ( t , x ) ) \mathcal {M}(u(t,x)) is contained in the frequency module of f f .