The Ryukyu Islands are a long archipelago off the main islands of Japan. Settled at an early period, the Ryukyu's independence from Japan is proven by the linguistic difference between the Japanese and Okinawans. Though written with many of the same characters, Japanese and the several languages of the Ryukyus are very different linguistically (and unintelligible to speakers of the other language) showing that the islands had been isolated from contact with Japan for a very long period of time. Japan reasserted control of the Okinawan chain after the Meiji Restoration in the 1860s, and today the traditional Ryukyun languages have largely died out.
Okinawa has an interesting philatelic history. As part of Japan, the islands issued no stamps of their own and used traditional Japanese stamps in much the same way that the Florida Keys issue no stamps of their own and use US stamps. The islands were not sovereign. During WWII, the Americans invaded Okinawa in April of 1945 and after a long and bloody battle finally defeated the Japanese in June and had a firm foothold towards the main islands of Japan. It was the bloodiness and fierceness of the battle for Okinawa that both convinced the Allies to use the Atom Bomb in August and the Japanese to surrender. The Americans saw the kind of casualties that further battles would cause, and the Japanese saw the inevitability of defeat. Because of the American casualties in Okinawa and because of the large military bases that we established there, America retained sovereignty over the Ryukyu Islands after WWII.
Ryukyu Islands stamps are listed in the Scott US Specialized catalog as a US Possession, even though they are unusual in this regard (and in fact, it wasn't until about 1980, after many meetings and much discussion that the head of the Ryukyu collector/dealer establishment, John Head, convinced the Scott editors to put Ryukyu stamps in the Scott Specialized). The designs and appeal of the other Possessions are quite different from the Ryukyus. Canal Zone, after a few overprinted Panama issues, quickly resorted to overprinting US stamps and then to designs that feel like US postal issues (in design terms this is called imitating the "trade dress" of US stamps), using the same sizes, lettering and design outline. A collector who never saw a Canal Zone stamp before would quickly realize that he is likely seeing something from the US sphere of influence. The stamps of the Philippines, another US Possession, followed the same pattern. Guam has only US overprinted issues. The final US Possession, Hawaii, has quite different postage stamps from the United States, but this is because, though Hawaii is collected as a possession, it really was an independent nation. The Ryukyu Islands stamps though are of an entirely different character. Though a US possession, they look very similar to the stamps of Japan and were designed and printed in Japan.
The Ryukyu Islands were given back to Japan in 1972 (because this was done by a Republican President, Nixon, it produced far less jingoistic outcry than was the case when President Carter negotiated the return of Canal Zone sovereignty to Panama eight years later in 1980. These have always been a double standard on the Right about how tough we need to appear in foreign relations). As a "dual country", one that has enjoyed ownership by two countries, the Ryukyus should be very popular. If Americans and Japanese both want to collect these stamps, with a combined population of 450 million of the wealthiest and most educated people in the world, you would expect Ryukyu stamps to sell briskly. This is not the case. In America, perhaps because we have no great pride in our history of occupying other nations, collectors have never cottoned much to our Possession issues. They have always been a poor stepchild to mainstream US philately, garnering few collectors. Further, the design types of Ryukyun stamps seem too Japanese for most American collectors. And the Japanese seem to love collecting the stamps of places they have occupied but are less fond of the philately that shows things the other way around. Ryuku Islands are a very completable and interesting philatelic specialty worthy of your attention if your are thinking of broadening your philatelic sphere.