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Continued Fractions and the Modular Surface

Luis B. Castro
(Institut for Matematik)
Foredrag for studerende
Fredag, 2 oktober, 2015, at 15:15-16:00, in Aud. D4 (1531-219)
Abstrakt:
A continued fraction is a formal expression
\[ \left[ a_0; a_1, a_2, \dots \right] = a_0 + \cfrac{ 1 }{ a_1 + \cfrac{ 1 }{ a_2 + \ddots } }, \]
where $ a_0 \in \mathbb{Z} $ and $ a_n \in \mathbb{N} $ for all $ n \ge 1 $.
There is a one to one correspondence with continued fractions and real numbers
But there is also a correspondence between continued fractions and endpoints of oriented geodesics on the modular surface $ M = \mathbb{H} / PSL(2,\mathbb{Z}) $ where $ \mathbb{H} $ is the hyperbolic plane.

If we look at $ \mathbb{H} $ we can tesselate it by the Farey tesselation $ \mathbb{F} $ which consists of triangles.
If we then have an oriented geodesic $ \overline{ \gamma } $ on $ M $ we can lift it to a geodesic $ \gamma $ on $ \mathbb{H} $.
We can define a cutting sequence $ \dots L^{ n_{ -1 } } R^{ n_0} L^{ n_1 } \dots $.
This cutting sequence tells us how $ \gamma $ cuts the triangles of $ \mathbb{ F } $ but it turns out that it also inculdes information about the left and right endpoints ($ \gamma_{ - \infty } $ and $ \gamma_\infty $ respectively).
Namely
\[ \gamma_\infty = \left[ n_1; n_2, \dots \right], \quad \frac{ - 1 }{ \gamma_{ -\infty } } = \left[ n_0; n_{ -1 }, \dots \right]. \]
After having established this I will use this to prove that the continued fraction representation of a real number $ \alpha $ becomes periodic if and only if $ \alpha $ is a quadractic irrational i.e. $ \alpha $ is root of a 2nd degree polynomial with integer coefficients.
Kontaktperson: Mads Bech