  Given a spectral triple on a $C^*$-algebra $\mathcal A$ together with a
unital injective endomorphism $\alpha$, the problem of defining a suitable
crossed product $C^*$-algebra endowed with a spectral triple is addressed. The
proposed construction is mainly based on the works of Cuntz and of Hawkins,
Skalski, White and Zacharias, and on our previous papers. The embedding of
$\alpha(\mathcal A)$ in $\mathcal A$ can be considered as the dual form of a
covering projection between noncommutative spaces. A main assumption is the
expansiveness of the endomorphism, which takes the form of the local
isometricity of the covering projection and is expressed via the compatibility
of the Lip-norms on $\mathcal A$ and $\alpha(\mathcal A)$.
