2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101145We show that the Nielsen-Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum SU(n)-representations, for any fixed integer $n \geq 2$. In the Pseudo-Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) SU(2)-TQFT representation matrices. It follows that at big enough levels, Pseudo-Anosov mapping classes are represented by matrices of infinite order.13 pages, minor modifications, to be published in Math. Proc. Camb. Phil. SocGeometric TopologyQuantum Algebra57R56 (Primary) 57M50, 37E30 (Secondary)text