On the Behaviour of the spectral characteristic of Feigenbaum’s map

Abstract

Let T g : [−1, 1] → [−1, 1] be the Feigenbaum map. It is well known that T g has a Cantor-type attractor F and a unique invariant measure µ0 supported on F. The corresponding unitary operator (U g φ)(x) = φ(g(x)) has pure point spectrum consisting of eigenvalues λ n,r , n ≥ 1, 0 ≤ r ≤ 2 n−1 − 1 with eigenfunctions e () (x). Suppose that f ∈ C 1([−1, 1]), f′ is absolutely continuous on [−1, 1] and f″ ∈ L p ([−1, 1], dµ0), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T g we prove that S n (f) ∼ 2−n q n , as n → ∞, where the constant q ∈ (0, 1) does not depend on f.