  We investigate one-electron properties of one-dimensional self-similar
structures called limit quasi-periodic lattices. The trace map of such a
lattice is nonconservative in contrast to the quasi-periodic case, and we can
determine the structure of its attractor. It allows us to obtain the three new
features of the present system: 1) The multi-fractal characters of the energy
spectra are {\it universal}. 2) The supports of the $f(\alpha)$-spectra extend
over the whole unit interval, $[0, 1]$. 3) There exist marginal critical
states.
