Exploring Higher Order Local Entropy Functions via Baire Category Theory

• Tingting Wang ^[1] ; Bilel Selmi ^[2] ; Zhiming Li ^[1]
  1. [1] Northwest University

     Northwest University

     China

     
  2. [2] University of Monastir

     University of Monastir

     Túnez

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
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• Resumen

  • 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.