ABSTRACT We prove that a periodic orbit P with coprime over-rotation pair is an over-twist periodic orbit iff the P-linear map has the over-rotation interval with left endpoint equal to the over-rotation number of P. We show that this fails if the over-rotation pair of P is not coprime. We give examples of patterns with non-coprime over-rotation pairs, no block structure over over-twists, and with over-rotation number equal to the left endpoint of the forced over-rotation interval (call them very badly ordered, similar to patterns of degree one circle maps in [L. Alseda, J. Llibre, and M. Misiurewicz, Badly ordered cycles of circle maps, Pacific J. Math. 184 (1998), pp. 23–41]). This presents a situation in which the results about over-rotation numbers on the interval and those about classical rotation numbers for circle degree one maps are different. In the end, we explicitly describe the strongest unimodal pattern that forces a given over-rotation interval and use it to construct unimodal very badly ordered patterns with arbitrary non-coprime over-rotation pairs.

中文翻译：

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间隔映射的非常糟糕的循环

摘要 我们证明了具有互质过旋转对的周期轨道 P 是一个过扭曲周期轨道，如果 P 线性映射具有左端点等于 P 的过旋转数的过旋转间隔。我们证明了这个如果 P 的过度旋转对不是互质的，则失败。我们给出了具有非互质过度旋转对、没有过度扭曲的块结构以及过度旋转数等于强制过度旋转间隔的左端点的模式示例（称它们为非常糟糕的有序，类似于模式[L. Alseda, J. Llibre, and M. Misiurewicz, Badlyordered cycle of circle maps, Pacific J. Math. 184 (1998), pp. 23–41] 中的一级圆图。这就出现了区间上的过度旋转数和圆度1映射的经典旋转数的结果不同的情况。最后，我们明确地描述了强制给定过度旋转间隔的最强单峰模式，并使用它来构造具有任意非互质过度旋转对的单峰非常糟糕的有序模式。