On determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators

TL;DR

This paper presents a method to explicitly construct hyperbolic operators in SL(2,z) with prescribed periods in geometric continued fractions and explores an algorithm for period determination in such operators.

Contribution

It introduces an explicit construction technique for hyperbolic operators with given continued fraction periods and investigates an algorithm for period finding in SL(2,z) operators.

Findings

Explicit construction of hyperbolic operators with prescribed periods

Experimental analysis of the period construction algorithm

Insights into the Gauss Reduction Theory for SL(2,z) operators

Abstract

For a given sequence of positive integers we make an explicit construction of a reduced hyperbolic operator in SL(2,z) with the sequence as a period of a geometric continued fraction in the sense of Klein. Further we experimentally study an algorithm to construct a period for an arbitrary operator of SL(2,z) (the Gauss Reduction Theory).