Spectral measures with arbitrary Hausdorff dimensions
Xin-Rong Dai, Qiyu Sun
TL;DR
This paper investigates the spectral properties of Riesz product measures on Cantor sets, demonstrating the existence of spectral measures with any Hausdorff dimension, including zero and one-dimensional cases.
Contribution
It introduces the existence of spectral measures with arbitrary Hausdorff dimensions, expanding understanding of spectral properties in fractal measures.
Findings
Existence of spectral measures with arbitrary Hausdorff dimensions
Construction of non-atomic zero-dimensional spectral measures
Construction of one-dimensional singular spectral measures
Abstract
In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral measures and one-dimensional singular spectral measures.
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Taxonomy
TopicsMathematical Analysis and Transform Methods · Mathematical Dynamics and Fractals · Topological and Geometric Data Analysis