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Light confinement in an optical fiber with a fractal photonic bandgap cladding was analyzed. The fractal cladding is a stratified structure of two alternating dielectric layers with high- and low-refractive index whose thicknesses vary according to a classic triadic fractal construction. Propagation characteristics were calculated solving the boundary problem at the dielectric interfaces of the stratified cladding by means of the transmission matrix method. Due to the finite number of layers, perfect light confinement can not be obtained. However, the fundamental mode suffers from losses lower than that ones of a conventional Bragg fiber with same material parameters and similar number of layers.