KAM THEOREM FOR REVERSIBLE MAPPING OF LOW SMOOTHNESS WITH APPLICATION

Assume the mapping A{x(1) = x + omega + y + f(x, y), (x, y) E T(d)x B(r0) y1 = y + g(x, y), is reversible with respect to G : (x, y) 7 -> (-x, y), and vertical bar f vertical bar Ct(TdxB(r0)) <= epsilon 0, vertical bar g vertical bar CE+d(TdxB(r0)) <= epsilon 0, where B(r0) := f vertical bar y vertical bar <= r0 : y E Rd}, P = 2d + 1 +mu with 0 < mu < 1. Then when epsilon 0 = epsilon 0(d) > 0 is small enough and omega is Diophantine, the map A possesses an invariant torus with rotational frequency omega. As an application of the obtained theorem, the Lagrange stability is proved for a class of reversible Duffing equation with finite smooth perturbation.