Eventually fixed points of endomorphisms of virtually free groups

We consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, EvFix(φ): points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of End(G), if φ is aperiodic, if EvFix(φ) is finitely generated and if EvFix(φ) is a normal subgroup. In the cases where EvFix(φ) is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.