2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139761An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: decide if the element commutes with a finite order element and in this case compute this element. The structure theorem on which this algorithm is based provides also a new solution to the conjugacy problem.Dynamical SystemsGeometric Topology37D20, 37E30, 37B10text