Orbit equivalence of global attractors of semilinear parabolic di erential equations

We consider global attractors Af of dissipative parabolic equations ut = uxx + f(x; u; ux) on the unit interval 0 x 1 with Neumann boundary conditions. A permutation f is de ned by the two orderings of the set of (hyperbolic) equilibrium solutions ut 0 according to their respective values at the two boundary points x = 0 and x = 1: We prove that two global attractors, Af and Ag, are globally C0 orbit equivalent, if their equilibrium permutations f and g coincide. In other words, some discrete information on the ordinary di erential equation boundary value problem ut 0 characterizes the attractor of the above partial di erential equation, globally, up to orbit preserving homeomorphisms.