Multifractal analysis of the power-2-decaying Gauss-like expansion

Each real number x∈[0,1] admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as x=∑i∈N2−(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→∞⁡1n∑j=1ndj(x) if the limit exists. We investigate the sizes of the level sets E(ξ)...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 542; no. 1; p. 128834
Main Author: Wang, Xue-Jiao
Format: Journal Article
Language:English
Published: Elsevier Inc 01.02.2025
Subjects:
Hausdorff dimension
Khintchine spectrum
Lyapunov spectrum
Khintchine spectrum
Hausdorff dimension
Lyapunov spectrum
ISSN:0022-247X
Online Access:Get full text
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Summary:Each real number x∈[0,1] admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as x=∑i∈N2−(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→∞⁡1n∑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→∞⁡1n∑j=1nlog⁡(dj(x))=ξ} and {x∈(0,1]:limn→∞⁡1n∑j=1n2dj(x)=ξ}.
ISSN:0022-247X
DOI:10.1016/j.jmaa.2024.128834