  We study the crossed product $C^*$-algebra associated to injective
endomorphisms, which turns out to be equivalent to study the crossed product by
the dilated autormorphism. We prove that the dilation of the Bernoulli
$p$-shift endomorphism is topologically free. As a consequence, we have a way
to twist any endomorphism of a $\D$-absorbing $C^*$-algebra into one whose
dilated automorphism is essentially free and have the same $K$-theory map than
the original one. This allows us to construct purely infinite crossed products
$C^*$-algebras with diverse ideal structures.
