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 mu(0) supported on F. The corresponding unitary operator (U-g phi)(x) = phi(g(x)) has pure point spectrum consisting of eigenvalues lambda(n,r) n >= 1, 0 1. Consider the sum of the amplitudes of the spectral measure of f: S-n(f) := Sigma(2n-1)(r=0) vertical bar rho((n))(r)vertical bar(2), rho((n))(r) = integral(1)(-1) f(x) d mu(0)(x). Using the thermodynamic formalism for T-g we prove that S-n(f) similar to 2(-n)q(n), as n ->infinity, where the constant q is an element of (0, 1) does not depend on f.