arXiv subjects

W. Patrick Hooper

Publications and source records attributed to W. Patrick Hooper.

22 records · Page 2Linked to original sources

arXiv2012

Renormalization of Polygon Exchange Maps arising from Corner Percolation

We describe a 2 parameter family of polygon exchange transformations parameterized by points in a square. Whenever the two parameters are irrational, the polygon exchange has periodic orbits of arbitrarily large period. We show that for almost all parameters, the polygon exchange map has the property that almost every point is periodic. However, there is a dense set of irrational parameters for which this fails. By choosing parameters carefully, the measure of non-periodic points can be made arbitrarily close to full measure. These results are powered by a notion of renormalization which holds in a more general setting. Namely, we consider a renormalization of tilings arising from the Corner Percolation Model.

math.DS↗

arXiv2011

Another Veech Triangle

We show that the triangle with angles Pi/12, Pi/3 and 7*Pi/12 has the lattice property and compute this triangle's Veech group.

math.GT↗

arXiv2009

Generalized staircases: recurrence and symmetry

We study infinite translation surfaces which are Z-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithusen, we provide examples of infinite non-arithmetic lattice surfaces, as well as surfaces with infinitely generated Veech groups.

math.DS↗

arXiv2008

Dynamics on an infinite surface with the lattice property

Dynamical systems on an infinite translation surface with the lattice property are studied. The geodesic flow on this surface is found to be recurrent in all but countably many rational directions. Hyperbolic elements of the affine automorphism group are found to be nonrecurrent, and a precise formula regarding their action on cylinders is proven. A deformation of the surface in the space of translation surfaces is found, which "behaves nicely" with the geodesic flow and action of the affine automorphism group.

math.DS↗