Markov partitions for the geodesic flow on compact Riemann surfaces of constant negative curvature

TL;DR

This paper provides an explicit and rigorous construction of Markov partitions for the geodesic flow on compact Riemann surfaces of constant negative curvature, enhancing understanding of hyperbolic dynamics.

Contribution

It offers a detailed, explicit construction of Markov partitions for a specific class of hyperbolic flows, improving on abstract existing methods.

Findings

Explicit forms of rectangles and local cross sections derived

Local product structure calculated in detail

Construction enhances understanding of hyperbolic flow dynamics

Abstract

It is well-known that hyperbolic flows admit Markov partitions of arbitrarily small size. However, the constructions of Markov partitions for general hyperbolic flows are very abstract and not easy to understand. To establish a more detailed understanding of Markov partitions, in this paper we consider the geodesic flow on Riemann surfaces of constant negative curvature. We provide a rigorous construction of Markov partitions for this hyperbolic flow with explicit forms of rectangles and local cross sections. The local product structure is also calculated in detail.