"Our earliest estimated date of contact isad 1150 for Fatu Hiva, South Marquesas. This is close to the date estimated by radiocarbon dating for settlement of that island group13, raising the intriguing possibility that, upon their arrival, Polynesian settlers encountered a small, already established, Native American population. It was on the island of Fatu Hiva—the easternmost island in equatorial Polynesia—that Thor Heyerdahl hypothesized that Native American and Polynesian individuals might have contacted one another, based on islanders’ legends stating that their forefathers had come from the east39. The Marquesas lie at the latitude of Ecuador, and wind- and current-based simulations indicate that they are the islands most likely to be reached from South America via the strong east-to-west currents and winds at these equatorial latitudes4,40,41.

We cannot discount an alternative explanation: a group of Polynesian people voyaged to northern South America and returned42 together with some Native American individuals, or with Native American admixture, as speculated in ref. 10. We have dated the contact event to the time when Polynesian explorers were, according to some studies, making their longest-range voyages (the century surrounding ad 1200)—a time when these studies suggest that the Polynesian settlers discovered all remaining island groups in the Pacific, from Hawaii to New Zealand to Rapa Nui13,38,42. The Tuamotu Archipelago, which lies at the centre of the Polynesian islands in which we found a Native American component, is known to have been a Polynesian voyaging hub, and according to simulations it is the second most likely location to be reached when voyaging from South America4. Further population genetics collaborations with these genetically understudied island populations are needed to resolve these alternative hypotheses."

By "single contact event" I think they mean that all the contacts between Polynesia and South America happened around the same time period and stopped thereafter for some reason. So the contact could have been groups going in either (or both) direction around that time period.