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. In this paper, we study dynamical systems on an infinite translation surface with the lattice property. We build the surface S1 by gluing together two polygonal parabolas as in figure 1. This surface is infinite in many respects; it has infinite genus, infinite area, and two cone points with infinite cone angle. This study is motivated by the study of closed, finite area translation surfaces with the lattice property, a property which has been found to have great significance for dynamics. For instance, Veech has shown the geodesic flow on a finite area translation surface with the lattice property satisfies a dichotomy. In every direction, the geodesic flow is either completely periodic (decomposes into periodic cylinders), or minimal and uniquely ergodic. See [Vee89] or [MT02]. The surface S1 may be obtained from some of Veech’s original examples by a limiting process. See section 2. Moreover, theorem 1 demonstrates that S1 has the lattice property. The geodesic flow on the surface supports a trichotomy. In directions of rational slope, the flow is either completely periodic or highly nonrecurrent, decomposing into strips. In irrational directions, the flow is recurrent, but the flow contains no closed trajectories and no saddle connections. See section 4. We study the dynamics of the action of the affine automorphism group in section 5. Our theorem governs the dynamics of hyperbolic elements, Ĥ , of the affine automorphism group of S1. The action of Ĥ on S1 is not recurrent. Nonetheless, cylinders are distributed evenly, in 2000 Mathematics Subject Classification. 37D40;37D50,37E99,32G15.