Thermodynamic formalism for the positive geodesic flow on the modular surface

TL;DR

This paper investigates the thermodynamic formalism of the positive geodesic flow on the modular surface, defining pressure, proving the variational principle, and exploring conditions for analyticity and uniqueness of equilibrium states.

Contribution

It introduces the thermodynamic formalism for this flow, establishing foundational results and conditions that were later expanded upon in subsequent research.

Findings

Defined pressure for the flow

Proved the variational principle

Identified conditions for real analyticity and unique equilibrium states

Abstract

In this note we study the thermodynamic formalism for the positive geodesic flow on the modular surface. We define the pressure and prove the variational principle. We also establish conditions for the the pressure to be real analytic and for the potentials to have unique equilibrium states. The results in this paper were largely superceded by "Phase transitions for suspension flows" by Iommi and Jordan arXiv:1202.0849