Hopf alternative

Let $ T $ be an invertible transformation of a Borel space $ X $ with a $ \sigma $- finite quasi-invariant measure (cf. also Invariant measure) $ m $. One can single out the following natural types of behaviour of $ T $ on a measurable subset $ A \subset X $:

1) $ A $ is invariant, i.e., $ TA = A $;

2) $ A $ is recurrent, i.e., for almost every point $ x \in A $ there is an $ n = n ( x ) > 0 $ such that $ T ^ {n} x \in A $;

3) $ A $ is wandering, i.e., all its translations are pairwise disjoint (so that almost every point $ x \in A $ never returns to $ A $ under the iterated action of $ T $). The transformation $ T $ is called ergodic (cf. Ergodicity) 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 \in \mathbf Z $, is the whole space $ X $.

If $ A $ is a wandering set, then for almost every $ x \in A $ the orbit $ \{ T ^ {n} x \} $ is an ergodic component of the action of $ T $, and the group $ \mathbf Z \cong \{ T ^ {n} \} $ 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 $( the 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 $ \mathbf Z $) the Hopf decomposition was introduced in [a1]. It can also be obtained for actions of $ \mathbf R $( called flows), [a6], [a7], or for actions of general countable groups, [a2]. See [a3] for general references on the Hopf decomposition.

An ergodic transformation (cf. also Ergodicity) is conservative unless the space $ ( X,m ) $ consists of a single dissipative orbit. If $ m ( X ) < \infty $, then any measure-preserving invertible transformation is conservative (the Poincaré recurrence theorem; cf. Poincaré return theorem).

E. Hopf [a4], [a5] showed that for a surface $ M $ of constant negative curvature, conservativity of the geodesic flow $ \{ T ^ {t} \} $ on the unit tangent bundle $ SM $( with respect to the Liouville invariant measure) implies its ergodicity, so that the geodesic flow is either ergodic and conservative or completely dissipative. The original proof of Hopf was based on the ratio ergodic theorem (cf. Ornstein–Chacon ergodic theorem) for conservative measure-preserving transformations [a1] and the fact that the distance between any two geodesics on the hyperbolic plane with the same end-point tends to zero (convergence of geodesics). Thus, the ratio Cesàro averages of any uniformly continuous function must coincide along any two geodesics which are asymptotic at $ + \infty $( or at $ - \infty $). Therefore, the ratio Cesàro averages are the same for all geodesics, whence $ \{ T ^ {t} \} $ is ergodic.

This argument easily carries over to more general situations. It can be simplified by using the usual Birkhoff ergodic theorem for induced transformations instead of the ratio ergodic theorem [a2].

Below, a modern formulation of the Hopf alternative for the geodesic flow on a Riemannian manifold $ M $ with pinched negative curvature is given, [a2]. The Hopf alternative can similarly be formulated for geodesic flows on $ { \mathop{\rm CAT} } ( - 1 ) $- spaces, trees and general Gromov hyperbolic spaces (cf. also Gromov hyperbolic space).

Invariant measures of the geodesic flow on $ SM $ are in one-to-one correspondence with measures on $ S {\widetilde{M} } $( $ {\widetilde{M} } $ denotes the universal covering space of $ M $) that are simultaneously invariant with respect to the geodesic flow and the action of the fundamental group $ G = \pi _ {1} ( M ) $. Since any infinite geodesic on $ {\widetilde{M} } $ is (up to parametrization) uniquely determined by the pair of its end-points on the sphere at infinity $ \partial {\widetilde{M} } $, there is a one-to-one correspondence between invariant Radon measures $ \Lambda $ of the geodesic flow on $ SM $ and $ G $- invariant Radon measures $ \lambda $ on $ \partial {\widetilde{M} } \times \partial {\widetilde{M} } \setminus { \mathop{\rm diag} } $( the latter measures are called geodesic currents). Denote by $ \Omega _ {r} \subset \partial {\widetilde{M} } $ the radial limit set of the group $ G $, see Hopf–Tsuji–Sullivan theorem.

Let $ \Lambda $ be a Radon-invariant measure of the geodesic flow, and let $ \lambda $ be the corresponding geodesic current. Suppose that $ \lambda $ is equivalent to a product of two measures on $ \partial {\widetilde{M} } $( this is the case for the Liouville-invariant measure of the geodesic flow as well as for other natural invariant measures). Then either:

a) the geodesic flow on $ M $ is conservative and ergodic with respect to the measure $ \Lambda $;

b) the action of $ G $ on $ \partial {\widetilde{M} } \times \partial {\widetilde{M} } \setminus { \mathop{\rm diag} } $ is ergodic and conservative with respect to the measure $ \lambda $;

c) the measure $ \lambda $ is concentrated on $ \Omega _ {r} \times \Omega _ {r} $; or:

a) the geodesic flow on $ M $ is completely dissipative with respect to the measure $ \Lambda $;

b) the action of $ G $ on $ \partial {\widetilde{M} } \times \partial {\widetilde{M} } \setminus { \mathop{\rm diag} } $ is completely dissipative with respect to the measure $ \lambda $;

d) the measure $ \lambda $ is concentrated on $ ( \partial {\widetilde{M} } \setminus \Omega _ {r} ) \times ( \partial {\widetilde{M} } \setminus \Omega _ {r} ) $.

References

[a1]	E. Hopf, "Ergodentheorie" , Springer (1937)
[a2]	V.A. Kaimanovich, "Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces" J. Reine Angew. Math. , 455 (1994) pp. 57–103
[a3]	U. Krengel, "Ergodic theorems" , de Gruyter (1985)
[a4]	E. Hopf, "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krummung" Ber. Verh. Sachs. Akad. Wiss. Leipzig , 91 (1939) pp. 261–304
[a5]	E. Hopf, "Ergodic theory and the geodesic flow on surfaces of constant negative curvature" Bull. Amer. Math. Soc. , 77 (1971) pp. 863–877
[a6]	U. Krengel, "Darstellungsätze für Strömungen und Halbströmungen I" Math. Ann. , 176 (1968) pp. 181–190
[a7]	U. Krengel, "Darstellungsätze für Strömungen und Halbströmungen II" Math. Ann. , 182 (1969) pp. 1–39