The structural and dynamic properties of low ionic strength micellar solutions of the cationic surfactant perfluorooctylbutane trimethylammonium bromide have been investigated by cryo-TEM, small-angle neutron scattering, small-angle X-ray scattering, T-jump and theological experiments. The surfactant molecules self-assemble into narrow ribbons with average dimensions on the order of 4 nm x 3 nm, either under salt-free conditions or in the presence of up to 30 mM KBr or NaF. Cryo-TEM also reveals in the salt-free systems the presence of networks of multiconnected micelles. Rheological experiments showed that these surfactant systems exhibit a strong shear-thickening effect even in the presence of up to 30 mM KBr. The T-jump response of the micellar solutions was found to be multiexponential. This observation rules out the presence of only linear micelles with an exponential length distribution and suggests more complex topologies of the micellar aggregates. The relaxation time associated with the predominant process in the T-jump relaxation is strongly correlated to the critical shear rate beyond which shear thickening occurs, thus indicating that this critical shear rate is controlled by the micellar kinetics.