Ratner’s Theorems and the Oppenheim Conjecture

Abstract

1 Introduction The field of dynamical systems studies the long-term behavior of a system that evolves under the repeated application of some transformation. For example, consider the solar system: its time evolution is approximated by Newtonian mechanics and laws of gravitation. The modern theory of dynamical systems originated at the end of the 19th century with an attempt to answer questions like, " What do the orbits in the solar system look like? (In particular, do they spiral into the Sun?) " The theory has since then developed into a broad field of mathematics with applications to meteorology, economics, astronomy, and other areas – including number theory. In this thesis, I will focus on a fairly recent set of powerful theorems in the theory of dynamical systems, proved by Marina Ratner around 1990. The theorems, in full generality, concern Lie groups and the actions of their subgroups generated by unipotent elements. They can be thought of as a sweeping generalization of the observation that a line on a 2-dimensional torus is either closed (that is, it wraps around the torus a finite number of times) or dense. Ratner's theorems assert that, in general, the closure of an orbit will be a very nice topological set. We prove a special case of these theorems for the Lie group SL 2 (R)/SL 2 (Z). We then present a surprising application of the theory of dynamical systems to number theory and the Oppenheim conjecture on the values of quadratic forms. The conjecture asserts that an indefinite quadratic form in n ≥ 3 variables is either proportional to a form defined over Z or its values on Z n are dense in R; the surprising connection between this number-theoretic result and the theory of dynamical systems was realized by M. S. Ranghunathan in the 1980s. The Oppenheim conjecture was first proved by G. A. Margulis, who gave a partial proof of the Ranghunathan conjectures in 1989. Ratner's work in the early 1990s proved the Ranghunathan and Margulis conjectures in full generality. The structure of this thesis is as follows. In Section 2 we present an introduction to the theory of dynamical systems and an overview of the key concept: ergodicity. In Section 3 we state some of Ratner's theorems on the orbits of dynamical systems under a unipotent flow; we then present a proof of the theorems for SL 2 (R)/SL 2 (Z), which …

Cite this paper

@inproceedings{Yudovina2008RatnersTA, title={Ratner’s Theorems and the Oppenheim Conjecture}, author={Elena Yudovina and Danijela Damjanovi{\'c}}, year={2008} }