2024/11/25 by Yubin He, He, Yubin
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2411.16045
openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate the Hausdorff measure of shrinking target sets in β-dynamical systems. These sets are dynamically defined in analogy to the classical theory of weighted and multiplicative approximation. While the Lebesgue measure and Hausdorff dimension theories for these sets are well-understood, the Hausdorff measure theory in even one-dimensional settings remains unknown. We show that the Hausdorff measure of these sets is either zero or full depending upon the convergence or divergence of a certain series, thus providing a rather complete measure theoretic description of these sets.