Purely infinite crossed products by endomorphisms

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 automorphism. 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.