This island issue is a territorial issue between Japan and South Korea and began after the conclusion of the San Francisco Peace Treaty in September 1951 (came into force on April 28, 1952, thereby officially ending the occupation of Japan by the US and their Allies after the Second World War); it was signed by Japan and the Allied Nations after the Second World War. Japan accordingly renounced its overseas territories (e.g., Taiwan, Manchuria, and Korea). South Korea (the Korean peninsula was divided after the Second World War, and South Korea represented both Koreas at the time) was not invited to the San Francisco Peace Conference because the Koreans did not fight against Japan.
South Korea submitted requests and laid claim to some islands; Takeshima/Dokdo was one of them. The US rejected the South Korean Takeshima claim since no records could be found that treated the island as part of Korea prior to its annexation by Japan (1910 – 1945). Takeshima had been under the jurisdiction of Shimane Prefecture, Japan, since about 1905; as a consequence, South Korea’s claim was rejected in August 1951.
However, the president of South Korea, Syngman Rhee, surprisingly and unilaterally declared South Korea’s maritime sovereignty ̶ known as the Syngman Rhee Line ̶ in January 1952, just three months before Japan’s sovereignty came into force. Needless to say, Takeshima Island and a large area of water was within the Line. South Korean authorities fired at a vessel of the Japan Coast Guard in July 1953 and again in August and November 1954. Since summer 1954, South Korea has enhanced its effective control over Takeshima Island.
Furthermore, up to 3911 Japanese fishermen (more than 40 lost their lives) and 327 ships were captured by South Korean authorities between 1947 and 1965. Many of these captures though occurred near Japanese archipelagos like Goto Islands or Tsushima Island, not Takeshima Island, according to recent researches. These fishermen were seized and detained for quite a long time; it was said then that this were hostage diplomacy. As the Treaty on Basic Relations between Japan and South Korea was concluded in 1965, South Korea might have misused the detained Japanese as hostages to maximize negotiation gains.
Ministry of Foreign Affairs of Japan.
Telegram 3470 to the Department of States from the U.S. embassy in Japan, April 27, 1960 by Douglas MacArthur, Wikisource. https://en.wikisource.org/wiki/Telegram_3470_to_the_Department_of_State
Dai Ichi Daihoumaru Ship case: Wikipedia. https://en.wikipedia.org/wiki/Dai_Ichi_Daihoumaru_Ship_case
National Diet Library, Japan: Dai 015 Kai Kokkai Suisan, Houmu, Gaimu Rengouiinkai
Dai 1 gou. http://kokkai.ndl.go.jp/SENTAKU/sangiin/015/1076/01502231076001c.html
Tony Marano Blog. http://propaganda-buster.blogspot.ch/search?q=takeshima
 Hiroki, Asaha: Nikkan Kankei Shi 2015, Tokyo University Press, p.249: in Li Shoban Lain to Nippongyosen daho, Kenji Fujii. http://www.ssri-j.com/SSRC/island/fujii-8-20170622.pdf#search=%27%E7%AB%B9%E5%B3%B6%E5%95%8F%E9%A1%8C+%E6%BC%81%E6%B0%91%E3%82%92%E4%BA%BA%E8%B3%AA%27
Akihiro, Iwashita: Hoppouryoudo, Takeshima, Senkaku, korega Kaiketsusaku, Asahishinsyo, Tokyo, 2013, p.31: in Li Shoban Lain to Nippongyosen daho, Kenji Fujii.