I'm not sure exactly what you mean by "space of subsets".
All I was trying to point out was that the argument that proves the recurrence theorem itself uses a volume of space around the initial state, and how "preimages" of that volume work. So it applies to volumes of states, too.
The Lorenz attractor generally avoids the recurrence because its dynamics are dissipative: nothing drives points near the attractor to points far from the attractor.
But once you are on the attractor, you can't just stay on the attractor forever getting "smeared out" without recurrence: you can only get at most smeared out over the finite area of the attractor, and eventually the smearing reaches your initial location on the attractor again.
All I was trying to point out was that the argument that proves the recurrence theorem itself uses a volume of space around the initial state, and how "preimages" of that volume work. So it applies to volumes of states, too.
The Lorenz attractor generally avoids the recurrence because its dynamics are dissipative: nothing drives points near the attractor to points far from the attractor.
But once you are on the attractor, you can't just stay on the attractor forever getting "smeared out" without recurrence: you can only get at most smeared out over the finite area of the attractor, and eventually the smearing reaches your initial location on the attractor again.