Notion of the renormalization group dynamical system, the self-avoiding fixed point and the critical trajectory are mathematically defined for the set of selfavoiding walks on the d-dimensional pre-Sierpin´ ski gaskets (n-simplex lattices), such that their existence imply the asymptotic behaviors of the self-avoiding walks, such as the existence of the limit distributions of the scaled path lengths of ‘‘canonical ensemble,’’ the connectivity constant (exponential growth of path numbers with respect to the length), and the exponent for mean square displacement. We apply the so defined framework to prove these asymptotic behaviors of the restricted self-avoiding walks on the 4-dimensional pre-Sierpin´ ski gasket.