Random walk and chaos of the spectrum. Solvable model

We consider the spectrum of the Laplacian corresponding to the random walk on the fractal graph depending on parameter β > 0. The spectrum of this Laplacian is given by the iteration of the polynomial R(β, x) = −(β + 2)x(x − 2) and the Julia set of this polynomial is the main part of the spectrum and it has a Cantorian nature. We prove that the Lebesgue measure of the spectrum is equal to zero for any β > 0. We consider the character of the spectrum for β → ∞ when the spectrum concentrates around two points 0, 2 and 1 is an isolated eigenvalue of infinite multiplicity. If β → 0 the spectrum σ(−Δ) approaches the segment [0, 2]. We prove that the spectral dimension ds is equal to and it converges to the Hausdorff dimension of the space for β