The starting point for this paper was the following problem: If we know the invariant densities for two maps S and T can we say something about the invariant density of the map $$U = T \circ S$$ U = T ∘ S ? This problem seems not to be touched on within ergodic theory. In this note we look at the inverse problem. Let the invariant density for $$U = T \circ S$$ U = T ∘ S be known. What can we say about invariant densities for S and T ? We discuss a simple model, namely the class of all piecewise fractional linear maps with two branches on the unit interval [0, 1].

中文翻译：

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组合分段分数线性映射的不变密度

这篇论文的出发点是以下问题：如果我们知道两个地图 S 和 T 的不变密度，我们能说一下地图 $$U = T \circ S$$ U = T ∘ S 的不变密度吗？这个问题似乎在遍历理论中没有涉及。在本笔记中，我们来看逆问题。让 $$U = T \circ S$$ U = T ∘ S 的不变密度是已知的。关于 S 和 T 的不变密度，我们能说些什么？我们讨论一个简单的模型，即在单位区间 [0, 1] 上具有两个分支的所有分段分数线性映射的类。