W. Patrick Hooper

The invariant measures of some infinite interval exchange maps

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions to which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than 1 . We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are of finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multitwists in a pair of cylinder decompositions in nonparallel directions. 37E05; 37E20, 37A40

An infinite surface with the lattice property I: Veech groups and coding geodesics

We study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface, and we show that this surface admits a deformation into other surfaces with topologically equivalent affine symmetries. The geodesics on these new surfaces are combinatorially the same as the geodesics on the

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 triangles Veech group.

Rel leaves of the Arnoux–Yoccoz surfaces

We analyze the rel leaves of the Arnoux–Yoccoz translation surfaces. We show that for any genus

Dynamics on the infinite staircase

mathbf {g}geqslant 3

Grid Graphs and Lattice Surfaces

g⩾3, the leaf is dense in the connected component of the stratum

Immersions and translation structures on the disk

{mathcal {H}}(mathbf {g}-1 ,mathbf {g}-1)