Cutting sequences for geodesic flow on the modular surface and continued fractions. (English) Zbl 1040.37008

Grabiner, David J.; Lagarias, Jeffrey C.

Monatsh. Math. 133, No. 4, 295-339 (2001).

This paper describes the cutting sequences of an geodesic flow on the modular surface. The set of cutting sequences for all geodesics is proved to form a two-sided shift that is not a sofic shift and which characterizes the standard fundamental domain of the hyperbolic plane. The authors establish strong relations among the cutting sequences associated with vertical geodesics, the Minkowski geodesic continued fraction expansion and the additive ordinary continued fraction expansion. They give explicit algorithms to deduce one from any other.

Reviewer: Anne Siegel (Rennes)

MSC:

37B10	Symbolic dynamics
11A55	Continued fractions
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37E15	Combinatorial dynamics (types of periodic orbits)

Keywords:

symbolic dynamics; cutting sequences; modular group; modular surface; continued fractions