  In this article, we study conjectures of Sandon on the minimal number of
translated points in the special case of the unit tangent bundle of a
Riemannian manifold. We restrict ourselves to contactomorphisms of $SM$ that
lift diffeomorphisms of $M$ homotopic to identity. We prove that there exist
sequences $(p_n,t_n)$ where $p_n$ is a translated point of time-shift $t_n$
with $t_n\to+\infty$ for a large class of manifolds. We also prove Morse
estimates on the number of translated points in the case of Zoll Riemannian
manifolds.
