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

• Published in: P-adic numbers, ultrametric analysis, and applications ultrametric analysis, and applications, 2012-10, Vol.4 (4), p.259-270
• Main Authors: Dzhalilov, A., Karakaya, V., Şimşek, N.
• Format: Article
• Language: English
• Subjects: Algebra; Mathematics; Mathematics and Statistics
• ISSN: 2070-0466; 2070-0474
• DOI: 10.1134/S2070046612040024

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 µ 0 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 r ( n ) ( x ). Suppose that f ∈ C 1 ([−1, 1]), f ′ is absolutely continuous on [−1, 1] and f ″ ∈ L p ([−1, 1], d µ 0 ), p > 1. Consider the sum of the amplitudes of the spectral measure of f : 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 .