Purely infinite crossed products by endomorphisms

We study the crossed product $C^{\ast }$-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 $\text{D}$-absorbing $C^{\ast }$-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 $C^{\ast }$-algebras with diverse ideal structures.