6 papers
Hausdorff dimension of the set of real numbers of Lagrange value three
Thomas A. Schmidt, Mark Sheingorn
We show that the set of real numbers of Lagrange value 3 has Hausdorff dimension zero by showing the appropriate generalization for each element of the Teichmueller space of the ap…
Individual Closed Horocyclic Orbits on the Modular Surface
Marvin Knopp, Mark Sheingorn
We track the trajectories of individual horocycles on the modular surface. Our tracking is constructive, and we thus \emph{effectively} establish topological transitivity and even…
Individual Horocyclic Orbits on , Closed and Otherwise
Marvin Knopp, Mark Sheingorn
This paper endeavors to track the trajectories of individual horocycles on \modsurf. It is far more common to study \emph{sets} of such trajectories, seeking some asymptotic behavi…
Horocyclic Orbits on $Γ(1)\frontslash\mathcal{H}$, \ Closed and Otherwise
Marvin Knopp, Mark Sheingorn
This paper studies certain horocyclic orbits on $Γ(1)\frontslash\mathcal{H}$. In the first instance we examine horocycles defined using the pencil of circles whose common point (in…
McShane's identity, using elliptic elements
Thomas A. Schmidt, Mark Sheingorn
We introduce a new method to establish McShane's Identity, based upon the fact that elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-p…
Low height geodesics and the Markoff spectrum
Thomas A. Schmidt, Mark Sheingorn
We classify, in terms of topology of highest arcs, low height non-simple geodesics on the modular hyperbolic punctured sphere with three elliptic fixed points of order two. Of eigh…