Partner orbits and action differences on compact factors of the hyperbolic plane. II: Higher-order encounters

Highlights: • A coordinatization for Poincaré sections found. • Mathematically rigorous definitions of 'encounters', 'partner orbits', 'orbit pairs', etc. given. • A periodic orbit including an $L$-parallel encounter has $(L-1)!-1$ partner orbits. • An inductive argument to construct partner orbits and estimate their action differences derived. • The action differences were approximated by terms of coordinates of the piercing points with an error bound. Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. We specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. The companion paper (Huynh and Kunze, 2015) proved the existence of a unique periodic partner orbit for a given periodic orbit with a small-angle self-crossing in configuration space that is a 2-encounter and derived an estimate for the action difference of the orbit pair. In this paper, we provide an inductive argument to deal with higher-order encounters: we prove that a given periodic orbit including an $L$-parallel encounter has $(L-1)!-1$ partner orbits; we construct partner orbits and give estimates for the action differences between orbit pairs.