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In this article, we study conjectures of Sandon on the minimal number oftranslated points in the special case of the unit tangent bundle of aRiemannian manifold. We restrict ourselves to contactomorphisms of $SM$ thatlift diffeomorphisms of $M$ homotopic to identity. We prove that there existsequences $(p_n,t_n)$ 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 Morseestimates on the number of translated points in the case of Zoll Riemannianmanifolds.

Morse estimates for translated points on unit tangent bundles