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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 coincides 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).

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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 coincides 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).