复多项式的重构定理
Reconstruction theorem for complex polynomials
摘要：
最近，Fornæss和Peters \cite{FP}在复分析环境中研究了Takens重构定理。他们研究了复多项式的实轨道，并证明了非例外多项式的遍历性质，如测度论熵，可以转移到实轨道映射上。在这里，我们表明，从\cite{FP}的结果也适用于例外的多项式，除非朱利亚集是完全包含在一个不变的垂直线，在这种情况下的熵是$0$。在\cite{T2} Takens证明了重建定理的自同态。在这种情况下，重建图不一定是嵌入，但是重建图的信息足以恢复原始图的$2m+1$ -st图像。我们的主要结果显示了一般复多项式的迭代和投影到实轴上的类似陈述。
英文摘要：
Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters \cite{FP}. They studied the real orbits of complex polynomials, and proved that for non-exceptional polynomials ergodic properties such as measure theoretic entropy are carried over to the real orbits mapping. Here we show that the result from \cite{FP} also holds for exceptional polynomials, unless the Julia set is entirely contained in an invariant vertical line, in which case the entropy is $0$. In \cite{T2} Takens proved a reconstruction theorem for endomorphisms. In this case the reconstruction map is not necessarily an embedding, but the information of the reconstruction map is sufficient to recover the $2m+1$-st image of the original map. Our main result shows an analogous statement for the iteration of generic complex polynomials and the projection onto the real axis.
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