Boundary maps and their natural extensions associated with Fuchsian and Kleinian groups
Open Access
- Author:
- Zydney, Adam J
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- February 23, 2018
- Committee Members:
- Svetlana Katok, Dissertation Advisor/Co-Advisor
Svetlana Katok, Committee Chair/Co-Chair
Federico Juan Rodriguez Hertz, Committee Member
Antole Katok, Committee Member
Kyusun Choi, Outside Member - Keywords:
- geodesic flow
symbolic dynamics
continued fractions
complex numbers
reduction theory
Fuchsian groups - Abstract:
- Geodesic flows on surfaces of constant negative curvature are a rich source of examples in ergodic theory, and geodesic flow on the modular surface in particular has deep connections to real continued fractions from number theory. This thesis deals with two extensions of this setting: either replacing the modular group PSL(2,Z) with a cocompact torsion-free Fuchsian group, or working with three-dimensional hyperbolic space and relating the boundary maps to continued fractions of complex numbers. The Fuchsian results are joint with Svetlana Katok and build on results of Katok and Ugarcovici, who studied a family of maps generalizing the Bowen-Series boundary map. When the parameters satisfy the short cycle property, i.e., the forward orbits at each discontinuity point coincide after one step, the natural extension map has a global attractor with finite rectangular structure. In this thesis, we generalize several results of Adler and Flatto, describing a conjugacy between the geometric and arithmetic maps and showing that the attractor parameterizes an associated arithmetic cross-section. This allows us to represent the geodesic flow as a special flow over a symbolic system. In cases where the cycle ends are discontinuity points, the resulting symbolic system is sofic. In the final chapter, we consider three-dimensional real hyperbolic space, in which the boundary is the Reimann sphere (C with a point at infinity) and the boundary maps use generators of Kleinian groups. Here the endpoints of geodesics can be described by complex continued fractions, which have been studied from the number theoretic perspective by Doug Hensley, S. G. Dani, and Arnaldo Nogueira, among others. In this thesis, a new "partition property" is described, substituting for the cycle property seen in the modular and Fuchsian literature. We state some results that apply to a wide range of boundary maps satisfying this partition property, and we discuss several specific algorithms in more detail. In many cases, the attractor of the natural extension map can be expressed as a finite union of products in C ⨯ C; this "finite product structure" is explicitly demonstrated for certain algorithms.