... was obviously not dancing with the New York City Ballet when he defected to the West (as implied, intentionally or otherwise, by this question). He was on tour with the Mariinsky Ballet. He initially joined the Royal Winnipeg Ballet, and from 1974 to 1978 he danced with the American Ballet Theatre, which is based in New York. In 1978 (according to Wikipedia) he abandoned his freelance career to spend 18 months as a principal of the New York City Ballet, run by the "legendary" George Balanchine.
Wikipedia also states that Baryshnikov became frustrated with dancing in the Soviet Union because, at only 5' 5", "he could not tower over a ballerina en pointe" and was restricted to secondary roles. Moreover, "the Soviet dance world [stuck] closely to 19th–century traditions and deliberately shunned the creative choreographers of the West, whose work Baryshnikov glimpsed in occasional tours and films."
He received an Oscar nomination for his part in The Turning Point, and in Sex and the City he played the final "love interest" of the central character, Carrie Bradshaw.
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