befamous或者bewell-known都是表示状态的，意思“有名的”表示过程，意思为“变得出名、变得有名”就用becomefamous根据语意来定。
to become famous
being famous
famous
（著名; 有名声） famous; well-known例如：He has gaind considerable repute.他已相当出名。Guilin is known for its beautiful sceneries.桂林因风景优美而出名。He is famous for his learning.他以有学问而出名。另外，词组如：make one's mark，rise to distinctiongain，(attain, win) distinction也是可以的。
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