I don't think the space of subsets is going to be bounded, which means the Poincaré recurrence theorem won't apply to it.
That coupled with the fact that starting our Lorenz equation on a box shaped subset will cause that box to stretch and distort, asymptotically approaching the Lorenz attractor. This implies that it can never become box shaped again. The set just gets closer and closer to the shape of the attractor as it evolves.
Otherwise it wouldn't really be an attractor now would it?
P.S. If you take a discrete subset, like pixels on the screen, then I agree those sets of pixels will reoccur, making it look like the box has reappeared. But in reality that is just no longer a representative sample of the twisted form of the actual set wrapped around the attractor.
I don't think the space of subsets is going to be bounded, which means the Poincaré recurrence theorem won't apply to it.
That coupled with the fact that starting our Lorenz equation on a box shaped subset will cause that box to stretch and distort, asymptotically approaching the Lorenz attractor. This implies that it can never become box shaped again. The set just gets closer and closer to the shape of the attractor as it evolves.
Otherwise it wouldn't really be an attractor now would it?
P.S. If you take a discrete subset, like pixels on the screen, then I agree those sets of pixels will reoccur, making it look like the box has reappeared. But in reality that is just no longer a representative sample of the twisted form of the actual set wrapped around the attractor.