[contravariant]

Your first requirement is part of the definition of a dynamical system. It is also satisfied for the evolution of the phase-space for any system in classical mechanics.

The second requirement is usually the case for systems with finite energy.

So as far as classical mechanics is concerned the Poincaré Recurrence Theorem pretty much always applies.

[cgadski]

Dynamical systems definitely don't have to preserve volume!

[contravariant]

Then your definition must be different from mine.

Which is possible. I mostly encountered them in the context of Ergodic Theory, in which case a preserved measure is very much non-optional.