Density spectrum of Cantor measure. (1st December 2022)

• Record Type: Journal Article
• Authors: Allaart, Pieter; Kong, Derong
• Is Part Of: Nonlinearity. Volume 35:Number 12(2022)
• Page Start: 6453
• Page End: 6484
• Publication Date: 2022-12-01
• Subjects: pointwise density -- Cantor measure -- density spectrum -- level set -- Hausdorff dimension -- entropy -- doubling map; pointwise density -- Cantor measure -- density spectrum -- level set -- Hausdorff dimension -- entropy -- doubling map; 28A80 -- 28A78 -- 37B10 -- 26A30; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac9a30
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Given ρ ∈ (0, 1/3], let μ be the Cantor measure satisfying μ = 1 2 μ f 0 − 1 + 1 2 μ f 1 − 1, where f i ( x ) = ρx + i (1 − ρ ) for i = 0, 1. The support of μ is a Cantor set C generated by the iterated function system { f 0, f 1 }. Continuing the work of Feng et al (2000) on the pointwise lower and upper densities Θ * s ( μ, x ) = lim inf r → 0 μ ( B ( x, r ) ) ( 2 r ) s, Θ * s ( μ, x ) = lim sup r → 0 μ ( B ( x, r ) ) ( 2 r ) s, where s = −log 2/log ρ is the Hausdorff dimension of C, we give a complete description of the sets D * and D * consisting of all possible values of the lower and upper densities, respectively. We show that both sets contain infinitely many isolated and infinitely many accumulation points, and they have the same Hausdorff dimension as the Cantor set C . Furthermore, we compute the Hausdorff dimension of the level sets of the lower and upper densities. Our method consists in formulating an equivalent 'dyadic' version of the problem involving the doubling map on [0, 1), which we solve by using known results on the entropy of a certain open dynamical system and the notion of tuning.