2024/04/26 by Xuejiao Wang, Wang, Xue-Jiao
Economics, Econometrics and Finance · #Complex Systems and Time Series Analysis #Dynamical Systems (math.DS) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2404.17414
openalex publication_date 2024/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Each real number x∈[0,1] admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as x=∑i∈ℕ 2-(d1(x)+d2(x)+⋯+di(x)), where di(x)∈ℕ. For any x∈(0,1], the Khintchine exponent γ(x) is defined by γ(x):=limn→∞(1)/(n)∑j=1ndj(x) if the limit exists. We investigate the sizes of the level sets E(ξ):=\x∈(0,1]:γ(x)=ξ\ for ξ≥ 1. Utilizing the Ruelle operator theory, we obtain the Khintchine spectrum ξ↦dimH E(ξ), where dimH denotes the Hausdorff dimension. We establish the remarkable fact that the Khintchine spectrum has exactly one inflection point, which was never proved for the corresponding spectrum in continued fractions. As a direct consequence, we also obtain the Lyapunov spectrum. Furthermore, we find the Hausdorff dimensions of the level sets \x∈(0,1]:limn→∞(1)/(n)∑j=1nlog(dj(x))=ξ\ and \x∈(0,1]:limn→∞(1)/(n)∑j=1n2dj(x)=ξ\.