method for the cationic polymerization of at least one olefinic monomer and more particularly the homopolymerization of a mono - olefinic monomer such as isobutylene and the copolymerization thereof with a di - olefinic monomer such as butadiene and optionally another mono - olefinic monomer in the presence of , as initiator of a halide of tetravalent vanadium , titanium and zirconium , an activator and under the influence of visible ultraviolet or infra - red light .