The paper has a section on this, around the end of page 4, which is really interesting. The short version is: They compared their double-precision results to extremely high-precision Taylor expansions (with theoretical 70+ digit accuracy, and calculated in 100+ digit accuracy), and found that they matched to the accuracy you'd expect.

That doesn't guarantee that the orbits are perfectly periodic, I suppose, but it does suggest that the orbits are stable with respect to rounding errors up to those you get from using doubles.