# On certain orbits of geodesic flow and (a,b)-continued fractions

## TL;DR

This paper characterizes special geodesic orbits on the Modular surface using a two-parameter family of continued fractions, extending the connection between dynamics and number theory.

## Contribution

It introduces a new characterization of exceptional geodesic orbits via (a,b)-continued fractions, expanding the Dani correspondence framework.

## Key findings

- Identification of two types of exceptional orbits
- Extension of Dani correspondence to new continued fraction families
- Enhanced understanding of geodesic flow and Diophantine approximation

## Abstract

In this article, we characterize two kinds of exceptional orbits of the geodesic flow associated with the Modular surface in terms of a two-parameter family of continued fraction expansion of endpoints of the lifts to the hyperbolic plane of the corresponding geodesics. As a consequence, we obtain an extension of Dani correspondence between homogeneous dynamics and Diophantine approximation.