倍角半共轭

摘要：

Franks定理的一个简单的推论是：当连续映射$g$同伦于圆周上的倍角映射时，它与之共轭，证明了当这种共轭有一个不连通的点逆时，则圆周上的典型点有一个具有不可数连通分量的点逆.进一步地，在这种情况下，$g$的拓扑熵严格大于角倍化的拓扑熵，并且双偶性具有无界变化。对于具有$D > 2$的度$D$圆映射，一个类似的定理也成立.

英文摘要：

A simple consequence of a theorem of Franks says that whenever a continuous map, $g$, is homotopic to angle doubling on the circle it is semiconjugate to it. We show that when this semiconjugacy has one disconnected point inverse, then the typical point in the circle has a point inverse with uncountably many connected components. Further, in this case the topological entropy of $g$ is strictly larger than that of angle doubling, and the semiconjugacy has unbounded variation. An analogous theorem holds for degree-$D$ circle maps with $D > 2$.

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