Spectral form factor in the Hadamard-Gutzwiller model: orbit pairs contributing in the third order

TL;DR

This paper analyzes third-order orbit pairs in the spectral form factor of the Hadamard-Gutzwiller model, proving the existence of partner orbits and providing explicit action difference estimates and a new symbolic dynamics framework.

Contribution

It introduces a proof of partner orbits for specific orbit pairs with two 2-encounters and develops a novel symbolic dynamics approach for orbit pairs.

Findings

Partner orbits exist for orbit pairs with two 2-encounters.

Explicit estimates for action differences are provided.

A new symbolic dynamics method via conjugacy classes is introduced.

Abstract

In this paper we consider orbit pairs contributing in the third order of the spectral form factor in the Hadamard-Gutzwiller model. We prove that periodic orbits including two 2-encounters in certain structures have partner orbits. The action differences are estimated at $\ln(1+u_1s_1)(1+u_2s_2)$ with explicit error bounds, where $(u_1,s_1)$ and $(u_2,s_2)$ are the coordinates of the piercing points. A new symbolic dynamics for orbit pairs via conjugacy classes is also provided.