Hopf Alternative

Let T be an invertible transformation of a Borel space X with a-nite quasi-invariant measure m. One can single out the following natural types of behaviour of T on a measurable subset A X: 1) A is invariant, i.e. TA = A; 2) A is recurrent, i.e., for a.e. point x 2 A there is n = n(x) > 0 such that T n x 2 A; 3) A is wandering, i.e., all its translations are pairwise disjoint (so that a.e. point x 2 A never returns to A under the iterated action of T). The transformation T is called ergodic if there are no non-trivial invariant sets (i.e., such that both the set and its complement have non-zero measure), conservative if there are no non-trivial wandering sets, and completely dissipative if there exists a wandering set A (a \fundamental domain") such that the union of its translations T n A; n 2 Z is the whole space X. If A is a wandering set, then for a.e. x 2 A the orbit T n x is an ergodic component of the action of T, and the group Z = fT n g acts freely on this orbit (an orbit with these two properties is a dissipative orbit). Conversely, the restriction of T onto any measurable set consisting of dissipative orbits is completely dissipative. Hence, the space X admits a unique Hopf decomposition into the union of two T-invariant disjoint measurable sets C and D (conservative and dissipative parts of X, respectively) such that the restriction of the action onto C is conservative, and the restriction onto D (which is the union of all dissipative orbits) is completely dissipative. For invertible transformations (i.e., measure type preserving actions of the group Z) the Hopf decomposition was introduced in 1]. It can be also obtained for actions of R (called ows) 2], or for actions of general countable groups 3]. See 4] for general references on the Hopf decomposition. An ergodic transformation is conservative unless the space (X; m) consists of a single dissipative orbit. If m(X) < 1, then any measure preserving invertible transformation is conservative (the Poincar e recurrence theorem). Hopf 5], 6] showed that for a surface of constant negative curvature M conservativity of the geodesic ow fT t g on the unit tangent bundle SM (with respect to the Liouville invariant measure) implies its ergodicity, so that the …