2022/04/28 by Zhi-Yi Wu, Wu, Yu-Liang, Wu, Zhi-Yi
Mathematics · #28A80 #42C05 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2204.13549
openalex publication_date 2022/04/28 · openalex created_date 2022/05/01 · openalex updated_date 2026/07/28
It is known [Dai and Sun, J. Funct. Anal. 268 (2015), 2464--2477] that there exist spectral measures with arbitrary Hausdorff dimensions, and it is natural to pose the question of whether similar phenomena occur for other dimensions of spectral measures. In this paper, we first obtain the formulae of Assouad dimension and of lower dimension for a class of Moran measures in dimension one that is introduced by An and He [J. Funct. Anal. 266 (2014), 343--354]. Based on these results, we show the existence of spectral measures with arbitrary Assound dimensions dimA and lower dimensions dimL ranging from 0 to 1, including non-atomic zero-dimensional spectral measures and one-dimensional singular spectral measures, and prove that the two values may coincide. In fact, more is obtained that for any 0 ≤ t ≤ s ≤ r ≤ u≤ 1, there exists a spectral measure μ such that dimL μ=t, dimH μ=s, dimPμ=r~ and dimAμ=u, where dimH and dimP denote the Hausdorff dimension and packing dimension of the measure μ, respectively. This result improves and generalizes the result of Dai and Sun more simply and flexibly.