Diophantine approximation is a rational approximation to an irrational number, which has been investigated by using continued fractions. In the thesis, we deal with three topics related to Diophantine approximation and continued fractions.
The first topic is the Markoff and Lagrange spectrum associated with the Hecke group. The classical Markoff and Lagrange spectrum is associated with the modular group, which has been investigated by using the regular continued fraction. We consider the Markoff and Lagrange spectrum associated with H_4 and H_6. We use the Romik dynamical system to show that some results on the classical Markoff and Lagrange spectra appear in the Markoff and Lagrange spectra associated with the Hecke group.
The Second topic is the exponents of repetition of Sturmian words. The exponent of repetition of a Sturmian word gives the irrationality exponent of the Sturmian number associated with the Sturmian word. For an irrational number θ, we determine the minimum of the exponents of repetition of Sturmian words of slope θ. We also investigate the spectrum of the exponents of repetition of Sturmian words of the golden ratio.
The last topic is quasi-Sturmian colorings on regular trees. We characterize quasi-Sturmian colorings of regular trees by its quotient graph and its recurrence function. We obtain an induction algorithm of quasi-Sturmian colorings which is analogous to the continued fraction algorithm of Sturmian words.