We investigate one-electron properties of one-dimensional self-similar structures called limit quasiperiodic lattices. The trace map of such a lattice is nonconservative in contrast to the quasiperiodic 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 universal. (2) The supports of the f(agr)-spectra extend over the whole unit interval, [0, 1]. (3) There exist marginal critical states.