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We show that the group of homeomorphisms of the Cantor set  H (2 ℕ ) has ample generics, that is, for every  m  the diagonal conjugacy action  g ·( h  1 ,  h  2 , …,  h   m  )=( gh  1  g  −1 ,  gh  2  g  −1 , …,  gh   m   g  −1 ) of  H (2 ℕ ) on  H (2 ℕ )  m   has a comeager orbit. This answers a question of Kechris and Rosendal. We show that a generic tuple in  H (2 ℕ )  m   can be taken to be the limit of a certain projective Fraïssé family. We also give an example of a projective Fraïssé family, which has a simpler description than the one considered in the general case, and such that its limit is a homeomorphism of the Cantor set that has a comeager conjugacy class.

The group of homeomorphisms of the Cantor set has ample generics