This is how Wikipedia summarizes the Poincaré Recurrence Theorem: This is wrong, isn't it? Don't you need to ensure the phase space is bounded, and isn't conservation of energy an insufficient justification for that? Like, imagine throwing two baseballs away from each other into infinite space at escape velocity or higher; surely energy is conserved, yet they'll never come back together. That seems incredibly basic, so I apologize if I'm asking something really stupid here, but please check me on this. Elsewhere, I've seen the theorem presented like this: My question is this: How do you know if the dynamics are restricted to a bounded subset of phase space or not? What condition establishes that fact?