Simple continued fractions and cutting sequences

Abstract

The connection between cutting sequences of a directed geodesic in the tessellated hyperbolic plane ℍ2, the modular group Γ = PSL(2, ℤ) and the simple continued fractions of an end point w of the geodesic have been established by Series [13]. In this paper we represent the simple continued fractions of w ∈ ℝ and the “L” and “R” codes of the cutting sequence in terms of modular and extended modular transformations. We will define a T 0-path on a graph whose vertices are the set of Farey triangles, as the equivalent of the cutting sequence. The relationship between the directed geodesic with end point w on ℝ, the Farey tessellation and the simple continued fraction expansion of w ∈ ℝ ∞ then follows easily as a consequence of this redefinition. Finite, infinite and periodic simple continued fractions are subsequently examined in this light.