On symmetries of iterates of rational functions
On symmetries of iterates of rational functions
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关于有理函数迭代的对称性
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发表时间：
2020
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通讯作者：
F. Pakovich
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作者：
F. Pakovich
令$A$ 为$ngeq 2$ 次的有理函数。我们用$G(A)$表示莫比乌斯变换组$sigma$，使得$Acirc sigma=
u circ A$ 对于一些莫比乌斯变换 $
u$，以及 $Sigma(A)$ 和 ${
m Aut}(A)$ $ G(A)$ 的子群，由莫比乌斯变换 $sigma$ 组成，使得 $Acirc sigma= A$ 和 $Acirc sigma= sigma circ A$ 相应。我们证明，除非 $A$ 具有非常特殊的形式，否则群 $G(A^{circ k})$, $kgeq 1,$ 的阶数是有限的，并且仅以 $n$ 为界。我们还证明了一些结果，允许我们在某些情况下显式计算组 $Sigma_{infty}(A)=cup_{k=1}^{infty} Sigma(A^{circ k})$ 和 ${
m Aut}_{infty}(A)=cup_{k=1}^{infty} {
m Aut}(A^{circ k})$，从动力学角度来看特别有趣。此外，我们证明与 $A$ 共享迭代的 $d$ 有理函数 $B$ 的数量是有限的，并且仅以 $n$ 和 $d$ 为界。
Let $A$ be a rational function of degree $ngeq 2$. We denote by $ G(A)$ the group of Mobius transformations $sigma$ such that $ Acirc sigma=
u circ A$ for some Mobius transformations $
u$, and by $Sigma(A)$ and ${
m Aut}(A)$ subgroups of $ G(A)$, consisting of Mobius transformations $sigma$ such that $ Acirc sigma= A$ and $ Acirc sigma= sigma circ A$, correspondingly. We show that, unless $A$ has a very special form, the orders of the groups $ G(A^{circ k})$, $kgeq 1,$ are finite and uniformly bounded in terms of $n$ only. We also prove a number of results allowing us in some cases to calculate explicitly the groups $Sigma_{infty}(A)=cup_{k=1}^{infty} Sigma(A^{circ k})$ and ${
m Aut}_{infty}(A)=cup_{k=1}^{infty} {
m Aut}(A^{circ k})$, especially interesting from the dynamical perspective. In addition, we prove that the number of rational functions $B$ of degree $d$ sharing an iterate with $A$ is finite and bounded in terms of $n$ and $d$ only.