Instead of comparing the *Poincare recurrence theorem* with *ergodic theorems* one should rather look at the underlying notions of *conservativity* and *ergodicity* in the general context of a measure class preserving action of a countable group. If the Poincare theorem says precisely that actions with a finite invariant measure are conservative, it is somewhat misleading to identify ergodicity with presence of an ergodic theorem (as this is the case for a very limited class of actions only).

In terms of the *ergodic decomposition* of an action, ergodicity (of course) means that there is only one ergodic component which coincides with the whole action. On the other hand, conservativity means that there are no *discrete ergodic components* (i.e., ones which coincide just with action orbits) - indeed, any wandering set obviously gives rise to discrete ergodic components, and it is not hard to see that the converse is also true. From this point of view **ergodicity is a strengthening of conservativity**.

However, there are several classes of dynamical systems (actions), for which conservativity and ergodicity are equivalent, i.e., any system from this class is either ergodic or *completely dissipative* (all ergodic components are discrete $\equiv$ the whole state space is the union of translates of a certain ``fundamental domain"). This phenomenon is called *"Hopf dichotomy"*, the most famous example of which (precisely the one originally studied by Hopf) is the case of geodesic flows on negatively curved manifolds (and of the associated boundary actions).