  We give a criterion which allows to prove non-ergodicity for certain infinite
periodic billiards and directional flows on 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 $Z^2$-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 billiard flow is not ergodic and has
uncountably many ergodic components. As another application, we show that for
any recurrent 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 Poincare' maps of the above systems are
associated to non-regular Z-valued cocycles for interval exchange
transformations.
