Abstract
We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Fr\c{a}czek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces.
Highlights
In 1986 Kerchoff, Masur and Smillie showed that the the geodesic flow on a compact translation surface is ergodic in almost every direction [17]
The study of the dynamical properties on periodic infinite translation surfaces is in its infancy
The authors noticed that the ergodicity reduces to a classically studied question of ergodicity of cylinder flows over circle rotations initiated by Conze and studied by many authors, see [4] for a good survey
Summary
In 1986 Kerchoff, Masur and Smillie showed that the the geodesic flow on a compact translation surface is ergodic in almost every direction [17]. If the compact square-tiled translation surface M is in the stratum H(2), and the infinite translation surface Mis a Z unramified cover of M , the infinite surface is not ergodic in a.e. direction They showed a similar result for some Z2 extensions such as the periodic full occupancy wind-tree model. Hubert have recently announced the following related result [3]: for a given f : [0, 1) → R with integral 0, which is the finite sum of characteristic functions of intervals, for a set of full measure of interval exchange transformations T the skew product Tf (x, i) = (T x, i+f (x)) into the closed subgroup of R generated by the values of f (x) is ergodic Their methods of proof are close to ours
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