Closed characteristics on non-compact hypersurfaces in $${\mathbb {R}^{2n}}$$

Abstract

Viterbo demonstrated that any (2n − 1)-dimensional compact hypersurface $${M \subset (\mathbb {R}^{2n},\omega)}$$ of contact type has at least one closed characteristic. This result proved the Weinstein conjecture for the standard symplectic space ( $${\mathbb {R}^{2n}}$$ , ω). Various extensions of this theorem have been obtained since, all for compact hypersurfaces. In this paper we consider non-compact hypersurfaces $${\mathbb {R}^{2n}}$$ coming from mechanical Hamiltonians, and prove an analogue of Viterbo’s result. The main result provides a strong connection between the top half homology groups H i (M), i = n, . . . , 2n − 1, and the existence of closed characteristics in the non-compact case (including the compact case).