论文标题
熵的灵活性，用于分段扩展单峰地图
Flexibility of entropies for piecewise expanding unimodal maps
论文作者
论文摘要
我们研究了熵（拓扑和度量标准）的灵活性，以扩展单峰地图的分段。我们表明，该类拓扑和度量熵值的唯一限制是两者都是积极的，拓扑熵最多是$ \ log 2 $，并且根据变异原理，度量熵不大于拓扑熵。 为了更好地控制公制熵，我们主要与拓扑混合分段扩展偏斜帐篷地图合作，只有2个不同的斜率。对于这些地图，还有一个额外的限制，即拓扑熵大于$ \ frac {1} {2} {2} \ log2 $。 我们还提出了一个有趣的观察结果，即，对于偏斜帐篷，映射临界值所有迭代的衍生物的衍生物的倒数总和为零。这是对连接拓扑熵的Milnor-Thurston公式的概括和不同的解释。
We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most $\log 2$, and by the Variational Principle, the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is an additional restriction that the topological entropy is larger than $\frac{1}{2}\log2$. We also make the interesting observation that for skew tent maps the sum of reciprocals of derivatives of all iterates of the map at the critical value is zero. It is a generalization and a different interpretation of the Milnor-Thurston formula connecting the topological entropy and the kneading determinant for unimodal maps.
