  We say $x \in \{0,1,2 \}^{\NN}$ is a word with Sturmian erasures if for any
$a\in \{0,1,2 \}$ the word obtained erasing all $a$ in $x$ is a Sturmian word.
A large family of such words is given coding trajectories of balls in the game
of billiards in the cube. We prove that the monoid of morphisms mapping all
words with Sturmian erasures to words with Sturmian erasures is not finitely
generated.
