Bullshit. The Chinese could not possibly have mapped both North and South America in the 15th century. Their ships were small and barely seaworthy.
Although there is alot for which to question the map, you are wrong about the abilities of Chinese naval vessels at the time.
The travels of the famous Chinese admiral in the book "1421, The Year China Discovered America" are documented in and outside of China, though the western-most extent of his expeditions are the least-well recorded and some are speculative.
The problem (a profound problem) was the Chinese Emperor at the time of that famous admiral's expeditions. Due to a complex set of internal, domestic reasons that Emperor had the admiral's entire fleet destroyed and banned any further such expeditions. That imperial attitude, about China's need to investigate things foreign prevailed under every Emperor thereafter and made China a "hermit", leading to technological and cultural stagnation at home while Europe advanced.
It is not a stretch to realize that a different decision by that Emperor and his successors may have produced a China that was not in comparatively dis-advantageous positions when industrialized Europe and Japan came calling in the 19th century.
With all that said, it is a stretch to characterize the map as representing any "claim" on America, no matter what is determined about its authenticity. Certainly, in terms of active settlement, the most prolific earliest "foreign" settlers in North America were the Vikings and their descendents from Greenland.
Au contraire. The Chinese gov't built quite large vessels. Trading vessels throughout time and geography have varied a great deal in size. Roman era vessels on the India trading run were quite large; grain haulers were generally larger still; some of the largest were used to transport 200 ton columns and whatnot from Egypt to Rome.
"The Chinese could not possibly have mapped both North and South America in the 15th century. Their ships were small and barely seaworthy."
Actually they could have in the 15th century, but the emperor ordered the fleet destroyed later. You need to read the history of the eunuch courtesan.
I have no idea personally whether the Chinese discovered America. But historian Jared Diamond disagrees with you (as do a number of other historians) about Chinese ships. He writes:
"As of the year 1400, China had by far the best, the biggest, and the largest number of, ocean-going ships in the world. Between 1405 and 1432 the Chinese sent 7 ocean-going fleets, the so-called treasure fleets, out from China. Those fleets comprised hundreds of ships; they had total crews of 20,000 men; each of those ships dwarfed the tiny ships of Columbus; and those gigantic fleets sailed from China to Indonesia, to India, to Arabia, to the east coast of Africa, and down the east coast of Africa. It looked as if the Chinese were on the verge of rounding the Cape of Good Hope, coming up the west side of Africa, and colonizing Europe."
The ships in Zeng He's fleet were the largest wooden sailing vessels ever built. His largest treasure ship was over 400 feet in length.
Wrong. They had been trading along the African coast for some time with ships that were larger than anything in Europe at the time, and probably more seaworthy.