Within a mere three or four centuries between about 1300 and 900 BC, the Lapita culture spread 6000 km further to the east from the Bismarck Archipelago, until it reached as far as Tonga and Samoa. They then navigated thousands of miles in the open ocean to reach most all of the islands in the Pacific. Did they navigate randomly to discover these? Perhaps. But it's also possible that they knew where to look for islands. They could have followed birds. But they also could have simply used the utter regularity of Pacific island geography. In fact, they are one of the more regular features of the Earth, and their regularity extends not only to the islands, but also to the underlying ridges on the ocean floor. If Polynesians found a way to detect changes in the depth of the ocean floor (e.g. ocean coloration), then they could have used that to navigate those to the closest island. A quick glance at an aerial photograph of the Pacific suffices. This reduces the problem of finding islands (akin to finding a needle in a haystack, with islands such as Easter Island having an area of only 163 square km in an ocean of 165 million square km) to the problem of finding ridges on the ocean floor --the lines that underly the archipelagos. That means the chance of stumbling upon Easter Island by chance is one in a million. On a clear day, though, a person standing at sea level can see a little less than three miles across the ocean. That means the area from which an island would be visible with the naked eye is 81 square km more --not a huge difference. It seems unlikely that Polynesians got to do a million far-reaching ocean voyages before discovering the average small island, as Polynesians were never overly populous, these long-distance voyages do not seem to have been a very regular occurrence, judging from Polynesian lore, and the population of the various islands seems to have happened rather quickly. 

My theory that Polynesians navigated in search of long archipelagos on ridges makes a testable prediction: islands out of these ridges should not have been populated by them. Indeed, the Galapagos Islands, 8,000 square km in area, or more than 40 times as large as the Easter Island, but not found on a ridge archipelago, never had any permanent settlement prior to the arrival of Europeans. 

So how did these Polynesians find the long island chains? The patterns are easily visible in aerial photography, but how did the ancient Polynesians get their aerial views. Wikipedia ventures provides several possible mechanisms:

Polynesian navigators employed a whole range of techniques including use of the stars, the movement of ocean currents and wave patterns, the air and sea interference patterns caused by islands and atolls, the flight of birds, the winds and the weather.^[15]

Scientists think that long-distance Polynesian voyaging followed the seasonal paths of birds. There are some references in their oral traditions to the flight of birds and some say that there were range marks onshore pointing to distant islands in line with these flyways. A voyage from Tahiti, the Tuamotus or the Cook Islands to New Zealand might have followed the migration of the Long-tailed Cuckoo (Eudynamys taitensis) just as the voyage from Tahiti to Hawaiʻi would coincide with the track of the Pacific Golden Plover (Pluvialis fulva) and the Bristle-thighed Curlew (Numenius tahitiensis). It is also believed that Polynesians employed shore-sighting birds as did many seafaring peoples. One theory is that they would have taken a frigatebird(Fregata) with them. These birds refuse to land on the water as their feathers will become waterlogged making it impossible to fly. When the voyagers thought they were close to land they may have released the bird, which would either fly towards land or else return to the canoe.^[15]

It is likely that the Polynesians also used wave and swell formations to navigate. Many of the habitable areas of the Pacific Ocean are groups of islands (or atolls) in chains hundreds of kilometers long. Island chains have predictable effects on waves and on currents. Navigators who lived within a group of islands would learn the effect various islands had on their shape, direction, and motion and would have been able to correct their path in accordance with the changes they perceived. When they arrived in the vicinity of a chain of islands they were unfamiliar with, they may have been able to transfer their experience and deduce that they were nearing a group of islands. Once they had arrived fairly close to a destination island, they would have been able to pinpoint its location by sightings of land-based birds, certain cloud formations, as well as the reflections shallow water made on the undersides of clouds. 

Winds, weather, currents, swells, waves, clouds and birds. Any of these might do in principle. In practice, though, the effect of small islands on currents is not very long-range. The same seems to be true about wave height and direction and winds. Mentioning birds simply postpones the question to that of how do birds know how to navigate to the islands. The answer lies in water depth. Birds would be able to see the color changes that allow one to perceive the structure of the ocean floor ridges from aerial views, and perhaps so would reflections on clouds. So we have our answer: birds and clouds afforded the ancient Polynesian navigators the aerial photography they needed to find ocean ridges, which themselves predict the location of linear archipelagos. As for non-linear archipelagos...well, ancient Polynesians never found them.

It has not escaped my attention that this theory suggests an answer to the long-standing riddle of the Nazca lines.

As for the question of whether Polynesians had contact with South America, there seems little doubt, not only due to the genetic analyses cited below, but also because if they managed to reach the Easter Island, which is 24 km long, then they certainly must have navigated past it at one time or another and hit the American mainland, which is tens of thousands of km long.

Relevant evidence:

• Chicken of Polynesian origin dated to before Columbus' arrival in America was found in Chile, but see also:

Although this initial report suggested a Polynesian pre-Columbian origin a later report looking at the same specimens concluded:

A published, apparently pre-Columbian, Chilean specimen and six pre-European Polynesian specimens also cluster with the same European/Indian subcontinental/Southeast Asian sequences, providing no support for a Polynesian introduction of chickens to South America. In contrast, sequences from two archaeological sites on Easter Island group with an uncommon haplogroup from Indonesia, Japan, and China and may represent a genetic signature of an early Polynesian dispersal. Modeling of the potential marine carbon contribution to the Chilean archaeological specimen casts further doubt on claims for pre-Columbian chickens, and definitive proof will require further analyses of ancient DNA sequences and radiocarbon and stable isotope data from archaeological excavations within both Chile and Polynesia.^[7]

• Polynesians had sweet potato, believed to be of South American origin, before the arrival of the Europeans.
• A stone tool made with a stone from Hawaii was found in the Tuamotus, French Polynesia (more than 4,000 km away).
• Polynesian genes show female Taiwanese ancestors, male Melanesian (Papua New Guinea) ancestors, and a small percentage of female South American ancestors.
• Lapita pottery shows Polynesians sailed east through Melanesia, not Micronesia.
• Voyagers have recreated ancient Polynesian canoes and sailed thousands of miles in them.
• Thor Heyerdahl and a crew of five sailed a raft modeled after ancient Peruvian rafts from Peru to the Polynesia with no modern equipment.
• A huge stone sculpture (of an outrigger canoe?) on a hill facing the sea in the Paracas peninsula in Peru, 130 miles from the site of the Nazca lines, is best read from 200 miles out at sea.
• All it takes to cross an ocean is an icebox