# Non-ergodic \(\mathbb{Z}\)-periodic billiards and infinite translation surfaces

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## Abstract

We give a criterion which proves non-ergodicity for certain infinite periodic billiards and directional flows on \(\mathbb{Z}\)-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a \(\mathbb{Z}^{2}\)-periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding directional billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent \(\mathbb{Z}\)-cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skew-products which arise as Poincaré maps of the above systems are associated to non-regular \(\mathbb{Z}\)-valued cocycles for interval exchange transformations.

## Mathematics Subject Classification (2000)

37A40 37C40## Notes

### Acknowledgements

We would like to thank Vincent Delecroix, Giovanni Forni and Pascal Hubert for useful discussions and suggestions that helped us improve the paper and Artur Avila for suggesting the argument used in Sect. 8. We also thank J.-P. Conze, A. Eskin, P. Hooper, M. Lemańczyk, C. Matheus, B. Weiss for useful discussions and the referee for suggestions to improve the presentation.

The first author is partially supported by the Narodowe Centrum Nauki Grant DEC-2011/03/B/ST1/00407. The second author is currently supported by an RCUK Academic Fellowship and the EPSRC Grant EP/I019030/1, whose support is fully acknowledged.

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