Trajectories and Diophantine approximations
In this chapter, our setting is the Poincaré half-plane. Consider a non-elementary, geometrically finite Fuchsian group Γ (see Chap. I for the definitions) which contains a non-trivial translation. With these hypotheses, the surface S=Γ\ℍ admits finitely many cusps (see Sects. I.3 and I.4) (Fig. VII.1).
As in the previous chapters, we let π denote the projection from ℍ to S. In the first step, we study the excursions of a geodesic ray π([z,x)) into the cusp corresponding to the image of the restriction of π to a horodisk centered at the point ∞. Our purpose is to relate the frequency of these excursions to the way in which the real number x is approximated by the Γ-orbit of the point ∞.
In the second step, we restrict our attention to the modular group and rediscover, in the spirit of Chap. III, some classical results of the theory of Diophantine approximations.
KeywordsModular Group Conical Point Fuchsian Group Irrational Number Diophantine Approximation
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