Abstract
For the 'infinite staircase' square tiled surface we classify the Radon invariant measures for the straight line flow, obtaining an analogue of the celebrated Veech dichotomy for an infinite genus lattice surface. The ergodic Radon measures arise from Lebesgue measure on a one parameter family of deformations of the surface. The staircase is a $\mathbb{Z}$-cover of the torus, reducing the question to the well-studied cylinder map.
Highlights
Gutkin [G] and Veech [V1, V2] classified the invariant measures for a linear flow on a square tiled surface, showing that it satisfies the Veech dichotomy: the flow is uniquely ergodic in any direction with irrational slope and periodic in any rational direction
M is the completion of M with respect to the flat metric, and M M consists of four cone singularities with “infinite cone angle.”
We use results about cylinder maps to prove the following analogue of the Veech dichotomy: Theorem 1. (i) If α is rational of the form p/q with p, q coprime and p or q even, in direction of slope α, the surface M decomposes into an infinite number of periodic cylinders
Summary
Gutkin [G] and Veech [V1, V2] classified the invariant measures for a linear flow on a square tiled surface (of finite area), showing that it satisfies the Veech dichotomy: the flow is uniquely ergodic in any direction with irrational slope and periodic in any rational direction. We use results about cylinder maps (skew products over rotations) to prove the following analogue of the Veech dichotomy: Theorem 1. (i) If α is rational of the form p/q with p, q coprime and p or q even, in direction of slope α, the surface M decomposes into an infinite number of periodic cylinders.
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