The site percolation transition in random Sierpinski carpets is investigated by real space renormalization. The fixed point is not unique like in regular translationally invariant lattices, but depends on the number It of segmentation steps of the generation process of the fractal. It is shown that, for each scale invariance ratio n, the sequence of fixed points p(n,k) is increasing with k, and converges when k-->infinity toward a limit p(n) strictly less than 1. Moreover, in such scale invariant structures, the percolation threshold does not depend only on the scale invariance ration, but also on the scale. The sequence p(n,k) and p(n) are calculated for n = 4, 8, 16, 32, and 64, and for k = 1 to k = 11, and k = infinity. The corresponding thermal exponent sequence nu(n,k) is calculated for n = 8 and 16, and for k = 1 to k = 5, and k = infinity. Suggestions are made for an experimental test in physical self-similar structures.
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