ETD

• laplacian eigenfunctions
• modular surface
• suspension model of the geodesic flow
• transfer operator

This thesis investigates Maass cusp forms, eigenfunctions of the Laplacian on the modular surface, whose explicit analytic structure remains largely elusive. Building on Helgason’s harmonic analysis and Lewis–Zagier’s period-like functions, a geometric–dynamical framework is developed, showing that different analytic representations—integral transforms, period-like functions, and functional equations—naturally emerge from the geodesic flow via a suspension model. The associated transfer operator encodes the expanding direction of the flow, with eigenfunctions corresponding to period-like solutions. The thesis further demonstrates that the integral kernels appearing in boundary representations arise directly from the lifted operator, providing a dynamical explanation for their analytic role and establishing a precise correspondence between automorphic distributions and Laplacian eigenfunctions. This approach unifies geometric, dynamical, and spectral perspectives, clarifies the mechanisms underlying period-like functions, and offers new insight into the structure of Maass cusp forms, suggesting broader principles applicable to other surfaces.