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Mathematics > Dynamical Systems
Title: On iterates of rational functions with maximal number of critical values
(Submitted on 13 Jul 2021 (v1), last revised 31 Oct 2023 (this version, v4))
Abstract: Let $F$ be a rational function of one complex variable of degree $m\geq 2$. The function $F$ is called simple if for every $z\in \mathbb C\mathbb P^1$ the preimage $F^{-1}\{z\}$ contains at least $m-1$ points. We show that if $F$ is a simple rational function of degree $m\geq 4$ and $F^{\circ l} =G_r\circ G_{r-1}\circ \dots \circ G_1$, $l\geq 1$, is a decomposition of an iterate of $F$ into a composition of indecomposable rational functions, then $r=l$ and there exist M\"obius transformations $\mu_i,$ $1\leq i \leq r-1,$ such that $G_r=F\circ \mu_{r-1},$ $G_i=\mu_{i}^{-1}\circ F \circ \mu_{i-1},$ $1<i< r,$ and $G_1=\mu_{1}^{-1}\circ F$. As applications, we solve a number of problems in complex and arithmetic dynamics for "general" rational functions.
Submission history
From: Fedor Pakovich [view email][v1] Tue, 13 Jul 2021 10:10:43 GMT (22kb)
[v2] Tue, 10 Jan 2023 14:02:04 GMT (22kb)
[v3] Wed, 18 Oct 2023 15:49:27 GMT (27kb)
[v4] Tue, 31 Oct 2023 15:48:01 GMT (27kb)
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