You are correct about the Lorenz attractor being dissipative. I just assumed the chaos of the Lorenz attractor and, say a frictionless double pendulum, were th

You are correct about the Lorenz attractor being dissipative. I just assumed the chaos of the Lorenz attractor and, say a frictionless double pendulum, were the same phenomenon. However it seems they are, in fact, quite different. Even still > So it applies to volumes of states, too. I'm pretty sure this is false. As I mention in another comment, http://philsci-archive.pitt.edu/9838/1/recurrence.pdf http://philsci-archive.pitt.edu/9838/1/recurrence.pdf says "In [the case of the space of probability distributions over phase space], the reason that classical dynamics fails to abide by the linear recurrence theorem is that the distribution space is infinite-dimensional, even if phase space has finite volume: distributions can have structure on arbitrarily short scales. Yes the proof of the recurrence theorem uses volumes; but the result still only applies to individual points within the volume (or maybe sets of measure 0 at best).