Conformal Maps, Monodromy Transformations, and Non-reversible Hamiltonian Systems

According to Arnol’d and Sevryuk, a Hamiltonian vector field XH is said to be weakly reversible if φ∗XH = −XH for some germ φ of real analytic transformation with φ(0) = 0, while XH is reversible if additionally φ is an involution, i.e., φ = Id. One also says that α1, . . . , αn are non-resonant, if k · α ≡ k1α1 + · · · + knαn = 0 (3) for all integers kj with k = (k1, . . . , kn) = 0. The main purpose of this note is to show the existence of non-reversible real analytic Hamiltonian systems of non-resonant eigenvalues. We shall prove Theorem 1. For n ≥ 2 there exist non weakly reversible Hamiltonian vector fields XH of the form (1) for which α1 ·α2 < 0, and α1, . . . , αn are non-resonant. We should mention that any real analytic Hamiltonian on R can be put into the Birkhoff normal form, if it starts with a non-degenerate quadratic form; in particular, its corresponding Hamiltonian system is reversible. Also, all Hamiltonian systems (2) are reversible by some formal involution when their eigenvalues satisfy the above non-resonance condition. Arnol’d and Sevryuk [1] gave a Hamiltonian function on R with vanishing quadratic form, of which the corresponding Hamiltonian vector field is not reversible by any linear involution. See