Geometric structures associated with a contact metric $(\kappa ,\mu )$-space

We prove that any contact metric $(\kappa ,\mu )$-space $(M,\xi ,\varphi ,\eta ,g)$ admits a canonical paracontact metric structure which is compatible with the contact form $\eta$. We study such canonical paracontact structure, proving that it verifies a nullity condition and induces on the underlying contact manifold $(M,\eta )$ a sequence of compatible contact and paracontact metric structures verifying nullity conditions. The behavior of that sequence, related to the Boeckx invariant $I_M$ and to the bi-Legendrian structure of $(M,\xi ,\varphi ,\eta ,g)$, is then studied. Finally we are able to define a canonical Sasakian structure on any contact metric $(\kappa ,\mu )$-space whose Boexkx invariant satisfies $\mathrm{\mid }{I}_M\mathrm{\mid }>1$.