Level-spacing distribution of a fractal matrix

We diagonalize numerically a Fibonacci matrix with fractal Hilbert space structure of dimension $d_f=\mathrm{1.8316...}$ We show that the density of states is logarithmically normal while the corresponding level-statistics can be described as critical since the nearest-neighbor distribution function approaches the intermediate semi-Poisson curve. We find that the eigenvector amplitudes of this matrix are also critical lying between extended and localized.