Thesis

Symbolic Dynamics of Billiard Flow in Isosceles Triangles

 

Work in Progress

Billiard dynamics in fractal tables (with Robert Neimeyer)

 

Publications

  Equidistribution of Horocyclic Flows on Complete Hyperbolic Surfaces of Finite Area (with John H. Hubbard),
  In the Tradition of Ahlfors-Bers V., Contemp. Math., vol. 510, A.M.S., Providence, RI, 2010, pp. 129-150

Miscellaneous Notes

Very rough Note on Symbolic Dynamics of Billiard Trajectories on Double Regular Odd-gons. (Refers to ideas, figures and definitions from Smillie and Ulcigrai's paper dealing in detail with this problem on the regular octagon, and more sketchily for general regular 2n-gons)

  Masur's Theorem on Recurrent Quadratic Differentials

Coming Soon...

   More Introductory Material about Flows on Translation Surfaces

   and Dynamics in Moduli Spaces of Quadratic Differentials:

Guide to Notation

Preliminaries on Translation Surfaces, their Moduli Spaces and Interval Exchange Maps

No Saddle-Connections Implies Minimality

Saddle-Connection Directions are Dense in S¹

Saddle-Connection Holonomy Vectors are Discrete in $ℝ$²

Stablizer of a Translation Surface is Discrete in SL₂($ℝ$)

Stabilizer is Never Co-compact

The Veech Dichotomy

Smillie's "Closed Orbit" Characterization of Lattice Surfaces

No-Small Triangles and No Small Virtual Triangles Characterizations of Lattice Surfaces

Uniform Complete Periodicity and Uniform Complete Parabolicity Characterizations of Lattice Surface

(and more....)

Bibliography