On Cross Sections to the Geodesic and Horocycle Flows on Quotients of $\operatorname{SL}(2, \mathbb{R})$ by Hecke Triangle Groups $G_q$

TL;DR

This paper models cross sections for geodesic and horocycle flows on quotients of SL(2,R) by Hecke triangle groups, using continued fraction algorithms to analyze invariant measures and extensions.

Contribution

It introduces a new model for cross sections based on continued fractions related to Hecke triangle groups, extending previous work on horocycle flows.

Findings

Derived a natural extension and invariant measure for a G_q-Farey interval map

Connected continued fraction algorithms with flow cross sections on quotient spaces

Provided a framework for analyzing dynamics on SL(2,R)/G_q

Abstract

In this paper, we provide a model for cross sections to the geodesic and horocycle flows on $\operatorname{SL}(2, \mathbb{R})/G_q$ using an extension of a heuristic of P. Arnoux and A. Nogueira. Our starting point is a continued fraction algorithm related to the group $G_q$, and a cross section to the horocycle flow on $\operatorname{SL}(2, \mathbb{R})/G_q$ from a previous paper. As an application, we get the natural extension and invariant measure for a symmetric $G_q$-Farey interval map resulting from projectivizing the aforementioned continued fraction algorithm.