Repetition invariant geometric means

We introduce a class of repetition invariant geometric means and obtain corresponding contractive barycentric maps of integrable Borel probability measures on the Cartan-Hadamard Riemannian manifold of positive definite matrices. They retain most of the properties of the Cartan barycenter and lead to the conclusion that there are infinitely many distinct contractive barycentric maps. Inequalities from the derived geometric means including the Yamazaki inequality and unitarily invariant norm inequalities are presented.