L^q-spectrum of graph-directed self-similar measures that have overlaps and are essentially of finite type

2024/10/12 by Yuanyuan Xie, Xie, Yuanyuan
Computer Science · Mathematics · #FOS: Mathematics #Random Matrices and Applications #Spectral Theory (math.SP) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2410.09459

openalex publication_date 2024/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For self-similar measures with overlaps, closed formulas of the L^q-spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. 106 (2019), 56--103]. We extend the results of Ngai and the author \citeNgai-Xie₂019 to the graph-directed self-similar measures. For graph-directed self-similar measures satisfying the graph open set condition, the L^q-spectrum has been studied by Edgar and Mauldin \citeEdgar-Mauldin₁992. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition. For graph-directed self-similar measures μ on \R^d (d≥1), which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the L^q-spectrum of μ for q≥ 0, and prove the differentiability of the L^q-spectrum. This framework allows us to include graph-directed self-similar measures that are strongly connected and not strongly connected and those in higher dimension.

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