in the support of . This is not an equivalence since for a transformation which is not uniquely ergodic, but for which there is an ergodic measure with full support , for any other ergodic measure the measure is not ergodic for but its orbits are dense in the support. Explicit examples can be constructed with shift-invariant measures. Mixing A transformation of a probability measure space is said to be mixing for the measure if for any measurable sets the following holds: It is immediate that a mixing transformation is also ergodic (taking to be a -stable subset and its complement). The converse is not true, for example a rotation with irrational angle on the circle (which is ergodic per the examples above) is not mixing (for a sufficiently small interval its successive images will not intersect itself most of the time). Bernoulli shifts are mixing, and so is Arnold's cat map. This notion of mixing is sometimes called strong mixing, as opposed to weak mixing which means that Proper ergodicity The transformation is said to be properly ergodic if it does not have an orbit of full measure. In the discrete case this means that the measure is not supported on a finite orbit of . Definition for continuous-time dynamical systems The definition is essentially the same for continuous-time dynamical systems as for a single transformation. Let be a measurable space and for each , then such a system is given by a family of measurable functions from to itself, so that for any the relation holds (usually it is also asked that the orbit map from is also measurable). If is a probability measure on then we say that is -ergodic or is an ergodic measure for if each preserves and the following condition holds: For any , if for all we have then either or . Examples As in the discrete case the simplest example is that of a transitive action, for instance the action on the circle given by is ergodic for Lebesgue measure. An example with infinitely many orbits is given by the flow along an irrational slope on the torus: let and . Let ; then if this is ergodic for the Lebesgue measure. Ergodic flows Further examples of ergodic flows are: Billiards in convex Euclidean domains; the geodesic flow of a negatively curved Riemannian manifold of finite volume is ergodic (for the normalised