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We investigate one-electron properties of one-dimensional self-similarstructures called limit quasi-periodic lattices. The trace map of such alattice is nonconservative in contrast to the quasi-periodic case, and we candetermine the structure of its attractor. It allows us to obtain the three newfeatures of the present system: 1) The multi-fractal characters of the energyspectra are {\it universal}. 2) The supports of the $f(\alpha)$-spectra extendover the whole unit interval, $[0, 1]$. 3) There exist marginal criticalstates.Comment: 4 pages, 2figure

Universalities in one-electron properties of limit quasiperiodic lattices