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<p>In this article we study <i>r</i>-neutralized local entropy and derive some entropy formulas. For an ergodic hyperbolic measure of a smooth system, we show that the <i>r</i>-neutralized local entropy equals the Brin-Katok local entropy plus <i>r</i> times the pointwise dimension of the measure. We further establish the existence of ergodic measures that maximize the <i>r</i>-neutralized entropy for certain hyperbolic systems. Moreover, we construct a uniformly hyperbolic system, for which such measures are not unique. Finally, we present some rigidity results related to these ergodic measures.</p>

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On r-Neutralized Entropy: Entropy Formula and Existence of Measures Attaining the Supremum

  • 期刊论文

  • 发表日期：

    2025年3月5日

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  • Changguang Dong,
  • Qiujie Qiao

摘要

In this article we study r-neutralized local entropy and derive some entropy formulas. For an ergodic hyperbolic measure of a smooth system, we show that the r-neutralized local entropy equals the Brin-Katok local entropy plus r times the pointwise dimension of the measure. We further establish the existence of ergodic measures that maximize the r-neutralized entropy for certain hyperbolic systems. Moreover, we construct a uniformly hyperbolic system, for which such measures are not unique. Finally, we present some rigidity results related to these ergodic measures.