The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues [13] recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy $ h(a) $ of the tent map $ f_a $, as a function of the parameter $ a = \exp(h_{top}(f_a)) $, is continuous and strictly increasing on $ [\sqrt{2},2] $. In this short note, we extend the last result and characterize regularity of the function $ h $ precisely. We prove that $ h $ is $ \frac{1}{2} $-Hölder continuous on $ [\sqrt{2},2] $ and identify its best Hölder exponent on each subinterval of $ [\sqrt{2},2] $. On the other hand, parallel to a recent result on topological entropy of the quadratic family due to Dobbs and Mihalache [7], we give a formula of pointwise Hölder exponents of $ h $ at parameters chosen in an explicitly constructed set of full measure. This formula particularly implies that the derivative of $ h $ vanishes almost everywhere.

中文翻译：

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帐篷族的公平熵

Misiurewicz和Rodrigues引入了公平度量和公平熵的概念[13最近，并详细讨论了分段单调间隔图。特别是，他们表明帐篷映射$ f_a $的公平熵$ h（a）$是参数$ a = \ exp（h_ {top}（f_a））$的函数，并且是连续且严格增加的在$ [\ sqrt {2}，2] $上。在本简短说明中，我们扩展了最后的结果，并精确地描述了函数$ h $的规律性。我们证明$ h $是$ \ frac {1} {2} $-Hölder在$ [\ sqrt {2}，2] $上连续，并在$ [\ sqrt {2}的每个子间隔上确定其最佳Hölder指数， 2] $。另一方面，与Dobbs和Mihalache [7]，我们给出了在显式构造的完整度量集中选择的参数下$ h $的逐点Hölder指数的公式。该公式特别暗示$ h $的导数几乎在所有位置都消​​失。