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We tackle holey fibers in full vectorial terms. From Maxwell's equations, we derive the dispersion relations of the modes guided by an infinitely self-similar air hole lattice. We focus in particular on the fundamental mode (the so-called space filling mode), and show that previous numerical results based on vector methods are accurate, but scalar ones are not. We also find the field flow lines, intensity distribution in the cross section, and linear polarization ratio vs. wavelength.