论文标题
在帐篷家族的公平熵上
On fair entropy of the tent family
论文作者
论文摘要
Misiurewicz和Rodrigues最近引入了公平度量和公平熵的概念，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）$作为参数$ a = \ exp（h_ {top}（f_a））$的函数，在$ [\ sqrt {2}，2]，2] $上是连续而严格增加的。在简短的说明中，我们扩展了最后的结果，并精确地表征了功能$ h $的规律性。我们证明$ h $是$ \ frac {1} {2} $ - hölder在$ [\ sqrt {2}，2] $上连续连续识别$ [\ sqrt {2}}，2] $的每个子interval上的最佳Hölder指数。另一方面，与最近由于Dobbs和Mihalache所致的二次家族拓扑熵的结果并行，我们在明确构造的完整度量集中选择的参数下给出了$ h $ $ h $的指数的指数。该公式特别意味着$ h $的衍生产品几乎在任何地方都消失了。
The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues 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, 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.
