Markov teorisi ve PGL(2,Z) grubunun dış otomorfizmi

The Markov numbers are positive integers that arise from the solutions of a special Diophantine equation, called Markov equation. These numbers appear in the context of continued fractions and so Diophantine approximation.The Lagrange number of the real number α is defined as the supremum of real numbers l such that |α − p/q| < 1/lq2 for an infinite number of rationals p/q. The Lagrange spectrum is defined as the set of Lagrange numbers of α ∈ R\Q . Further, the solutions are related with the Lagrange-Markov spectrum which consists of those quadratic numbers which are badly approximable by rational numbers. We focus here on the set of irrationals α, called Markov irrationals, with L(α) < 3 such that continued fraction expansion of such numbers is in the form [...,a1,a2,a3,...] with ai ∈ {1, 2}. However, there is a fundamental involution Jimm of real line induced by the outer automorphism of the extended modular group PGL(2, Z). Its action on the real line, explicitly on continued fraction expansion, recently discovered. In this dissertation, with the aim of finding relations on various parts of this work, we consider to ask whether there is a classification related to Markov number or a special property the set of image of Markov irrationals under the involution Jimm. Markov numbers also appear in many parts of mathematics such as binary quadratic forms, hyperbolic geometry and the combinatorial of words and we investigated analogous problem in different topics.