In order to apply Poincarè recurrence it is the set of available points of the phase space that must be “compact” and this is likely the case if we assume that the total energy of the universe is finite.

The energy constrains the moments, but not the positions. If there is infinite space, the phase space is also infinite, even at constant energy.

Take two balls which start both at x=0, one with velocity v(0) = 1 and the other with velocity v(0) = −1, in an infinite line. They will continue to go away forever, no recurrence.

In order to apply Poincarè recurrence it is the set of available points of the phase space that must be “compact” and this is likely the case if we assume that the total energy of the universe is finite.

The energy constrains the moments, but not the positions. If there is infinite space, the phase space is also infinite, even at constant energy.

Take two balls which start both at x=0, one with velocity v(0) = 1 and the other with velocity v(0) = −1, in an infinite line. They will continue to go away forever, no recurrence.

Good point but gravity could be enough to keep the available positions in a bounded set

While of course it

could, current measurements suggest that it is not.