Symbolic Dynamics of the Weyl Chamber Flow
Abstract:
This thesis studies codings of orbits of Weyl chamber flows on symmetric spaces of non-compact type.</br> Let H be the hyperbolic plane with constant curvature −1 and Γ be a Fuchsian group of finite covolume. Let D be a Dirichlet domain of Γ on H. The main result shows that the set of cutting sequences of all geodesics in the sense of Morse with respect to the tessellation of H, formed by the sets gD, g ∈ Γ, is a topological Markov chain if and only if D does not have vertices in H.</br> Also, a background is provided for the study of the generalization of continued fractions to higher dimensions. So-called arithmetic Gauss coding of geodesics on H is described along with its relation with the minus continued fractions. H is a particular case of a symmetric space of non-compact type, H = SL<sub>2</sub>R/SO<sub>2</sub>R, and the geodesic flow on H implements the Weyl chamber flow on it. A generalization of the minus continued fractions was suspected by S. Katok and A. Katok to exist, which involves orbits of Weyl chamber flows on symmetric spaces of non-compact type SL<sub>n</sub>R/SO<sub>n</sub>R and their compactifications.