论文标题
关于理性功能的迭代对称性
On symmetries of iterates of rational functions
论文作者
论文摘要
令$ a $是学位$ n \ geq 2 $的合理功能。让我们用$ g（a）$表示Möbius变换$σ$的组，以便$ a \circcσ=ν_σ\ circ a $对于某些möbius变换$ν_σ$，以及$σ（a）$ and $ {$和$ {\ rm aut}（a aut}（a）$ g（a）相应地，A \Circσ=σ\ Circ a $。在本文中，我们研究了以上$ a $ $ $的序列的序列。特别是，我们表明，如果$ a $与$ z^{\ pm n}，$，那么$ g（a^{\ circ k}）$，$ k \ geq 2的订单，$是有限的，均为有限的，仅以$ n $ $ n $均匀限制。我们还证明了$σ_{\ infty}（a）= \ cup_ {k = 1}^{\ infty}σ（a^{\ circ k}）$和$ {\ rm aut} _ {\ rm aut} _ { aut}（a^{\ circ k}）$，从动态角度来看，这特别有趣。
Let $A$ be a rational function of degree $n\geq 2$. Let us denote by $ G(A)$ the group of Möbius transformations $σ$ such that $ A\circ σ=ν_σ \circ A$ for some Möbius transformations $ν_σ$, and by $Σ(A)$ and ${\rm Aut}(A)$ the subgroups of $ G(A)$ consisting of $σ$ such that $ A\circ σ= A$ and $ A\circ σ= σ\circ A$, correspondingly. In this paper, we study sequences of the above groups arising from iterating $A$. In particular, we show that if $A$ is not conjugate to $z^{\pm n},$ then the orders of the groups $ G(A^{\circ k})$, $k\geq 2,$ are finite and uniformly bounded in terms of $n$ only. We also prove a number of results about the groups $Σ_{\infty}(A)=\cup_{k=1}^{\infty} Σ(A^{\circ k})$ and ${\rm Aut}_{\infty}(A)=\cup_{k=1}^{\infty} {\rm Aut}(A^{\circ k})$, which are especially interesting from the dynamical perspective.
