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<p>In this paper, we investigate the average metric mean dimension of typical compact sets in continuous dynamical systems. We introduce the concept of average metric mean dimension, which captures the iterative nature of dynamical systems via general averaging schemes. We prove that, under mild conditions, the lower average metric mean dimension vanishes for a typical compact set. In contrast, the upper average metric mean dimension is shown to be bounded from below by a certain maximal dimension derived from local dynamical homogeneity. Furthermore, we establish that this dichotomy of average metric mean dimensions is robust under a broad class of averaging procedures, including Hölder and Cesàro averaging.</p>

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Average Metric Mean Dimensions of Typical Compact Sets

  • 期刊论文

  • 发表日期：

    2026年7月14日

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  • Mengyao Zhang,
  • Bilel Selmi,
  • Megha Pandey,
  • Zhiming Li

摘要

In this paper, we investigate the average metric mean dimension of typical compact sets in continuous dynamical systems. We introduce the concept of average metric mean dimension, which captures the iterative nature of dynamical systems via general averaging schemes. We prove that, under mild conditions, the lower average metric mean dimension vanishes for a typical compact set. In contrast, the upper average metric mean dimension is shown to be bounded from below by a certain maximal dimension derived from local dynamical homogeneity. Furthermore, we establish that this dichotomy of average metric mean dimensions is robust under a broad class of averaging procedures, including Hölder and Cesàro averaging.