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<p>In this study, we rigorously prove that for a typical measure, the local entropy function of a continuous flow does not exist. Specifically, the pattern demonstrated by a typical measure is incredibly complex and irregular, and even after employing widely accepted and significantly effective smoothing techniques, including higher-order Riesz-Hardy logarithmic averages and Cesàro averages, the local entropy function of the flow still does not exist. More precisely, the lower average local entropy function is zero, while the upper average local entropy function is infinity.</p>

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Exploring Higher Order Local Entropy Functions via Baire Category Theory

  • 期刊论文

  • 发表日期：

    2025年6月19日

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  • Tingting Wang,
  • Bilel Selmi,
  • Zhiming Li

摘要

In this study, we rigorously prove that for a typical measure, the local entropy function of a continuous flow does not exist. Specifically, the pattern demonstrated by a typical measure is incredibly complex and irregular, and even after employing widely accepted and significantly effective smoothing techniques, including higher-order Riesz-Hardy logarithmic averages and Cesàro averages, the local entropy function of the flow still does not exist. More precisely, the lower average local entropy function is zero, while the upper average local entropy function is infinity.