easter island (rapa nui) very much was colonised by polynesians
i don't remember if they colonised the mainland but there is evidence that they traded with the mainland nations
There are six claims of evidence to support contact.
The presence of Sweet Potatoes in Polynesia is a fairly solid one, assuming that the sweet potato is accurately believed to be an American plant. There is all but conclusive evidence to support the Pre-Columbian presence of the plant, but it is possible that it was spread by non-human means.
The other solid claim is that Peruvian mummies have been found to be preserved with the sap of a specific sort of tree that is only known to grow in the South Pacific. The specific example tested is dated to around AD 1200. This is the solidest bit of evidence, as any non-human means of transport would have resulted in the actual tree being present. The alternative theory is that the tree used to be present but died out.
Linguistically, there are two words -the one for
axe and the one for
sweet potato- that are very similar in South American languages and Polynesian ones. This could be evidence of contact, but could also be a coincidence. That the word for
sweet potato is one of the two (and the stronger connection) does support it as evidence, given that the sweet potato is also considered evidence of contact.
There are some trace genetic similarities between the peoples of South America and those of Polynesia, but they are too weak to suggest colonization. The possibility that these result from trade missions is a sound one, however.
The other two commonly cited evidences have been debunked. A few California tribes operated a similar type of canoe to that used by the Polynesians (and dissimilar to that of other Amerindian tribes), but there is too much evidence of them developing the things for it to have been a copy. It was once thought that the chickens used in both places were related, with one group getting them from the other, but DNA testing has proved this to be false. Polynesia got their chickens from Asia, while the Americas had a native variety.
In short, there is some evidence to suggest Pre-Columbian contact, but there's not enough to
prove it.