Symbolic Dynamics of the Weyl Chamber Flow
Open Access
- Author:
- Egorov, Arseni
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- August 16, 2011
- Committee Members:
- Svetlana Katok, Dissertation Advisor/Co-Advisor
Svetlana Katok, Committee Chair/Co-Chair
Anatole Katok, Committee Member
Federico Rodriguez Hertz, Committee Member
Gutti Jogesh Babu, Committee Member - Keywords:
- topological Markov chains
geodesic flow
Weyl chamber flow
geometric coding
arithmetic coding
symmetric spaces
continued fractions - 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 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.