Automorphic Forms and Reeb-Like Foliations on Three-Manifolds

TL;DR

This paper develops methods to generate dynamical systems on 3-manifolds using automorphic functions and Reeb foliations, providing explicit equations and exploring different geometric models.

Contribution

It introduces explicit differential equations for systems on hyperbolic 3-manifolds via automorphic functions and examines dynamical systems constructed through Reeb foliations.

Findings

Derived explicit differential equations for hyperbolic 3-manifolds

Unified dynamical systems across fundamental regions

Explored Reeb foliation-based system construction

Abstract

In this paper, we consider different ways of generating dynamical systems on 3-manifolds. We first derive explicit differential equations for dynamical systems defined on generic hyperbolic 3-manifolds by using automorphic function theory to uniformize the upper half-space model. It is achieved via the modification of the standard Poincare theta series to generate systems invariant within each individual fundamental region such that the solution trajectories match up on the appropriate sides after the identifications which generate a hyperbolic 3-manifold. Then we consider the gluing pattern in the conformal ball model. At the end we shall study the construction of dynamical systems by using the Reeb foliation.