Closed Geodesics on the Punctured Torus
We examine the behavior of geodesics on the punctured torus. We first consider simple geodesics and the results of . In particular, it can be shown that the lifts of these geodesics into the upper half plane are semicircles with diameter not exceeding 3, and that these diameters correspond to values in the Markoff Spectrum. The methods used in  highlight a deep connection between simple geodesics, continued fractions, and sequences obeying the Dickson rules. The obvious generalizations (sequences breaking these rules) correspond to geodesics which intersect themselves. We then develop examples of such non-simple geodesics, and use the methods of  and  to understand how these geodesics, too, tie into continued fractions and Markoff numbers.
Franklin and Marshall College Archives, Undergraduate Honors Thesis 2007
- F&M Theses Collection