A quantitative closing lemma and partner orbits on Riemannian manifolds with negative curvature*

A famous conjecture in the physics community asserts that the spectral fluctuation of a classically chaotic system should be universal and can be attributed to correlations in the action spectrum of periodic orbits. As it has been observed by Sieber and Richter (2001 Phys. Scr. T90 128–33; 2002 J. Phys. A 35 L613–9) in certain models like geodesic flows on surfaces, closed geodesics with small self-crossing angle and their partners (Sieber–Richter pairs) play an important role in this analysis. A rigorous treatment in the case of closed surfaces of constant negative curvature have been provided by Huynh and Kunze (2015 Nonlinearity 28 593–623). We generalise their analysis to manifolds of variable negative curvature and arbitrary dimensions. Results and ideas in this paper are also of interest in their own right and should provide useful information in investigating the fine structure of the length spectrum. Our study of partner orbits is based on a quantitative version of the closing lemma. As an application we study partner and pseudo-partner orbits for self-crossing closed geodesic.