We prove an upper bound for the polynomial entropy of continuous, piecewise monotone maps of the interval, according to the number of intervals of monotonicity of its iterates. We give examples that show that this inequality is sharp. As a direct consequence of this inequality, the polynomial entropy of monotone, continuous, interval maps is always less than or equal to one. We give examples where we can also obtain lower bounds. We also prove analogous inequality in terms of total variations of the iterates of these interval maps. Also, this inequality is sharp.

中文翻译：

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间隔图和圈数的多项式熵

根据迭代的单调性的区间数，我们证明了区间的连续，分段单调图的多项式熵的上限。我们给出的例子表明，这种不平等现象很严重。这种不等式的直接结果是，单调，连续，间隔图的多项式熵始终小于或等于1。我们给出了一些示例，在这些示例中我们还可以获得下界。我们还证明了这些间隔图的迭代总变化方面的类似不等式。而且，这种不平等现象非常严重。