The conjugacy problem is one of the important questions in iteration theory. As far as we know, for discontinuous strictly monotone maps there is no complete result. In this paper, we investigate the conjugacy problem of strictly monotone maps with only one jump discontinuity. We give some sufficient and necessary conditions for the conjugacy relationship. And we present some methods to construct all conjugacies. Furthermore, we present the conditions to guarantee $$C^1$$ C 1 smoothness of these conjugacies.

中文翻译：

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仅一跳不连续的严格单调映射的共轭问题

共轭问题是迭代理论中的重要问题之一。据我们所知，对于不连续的严格单调映射，没有完整的结果。在本文中，我们研究了只有一个跳跃不连续性的严格单调映射的共轭问题。我们给出了共轭关系的一些充分必要条件。并且我们提出了一些构造所有共轭的方法。此外，我们提出了保证这些共轭的 $$C^1$$C 1 平滑度的条件。