On iterates of rational functions with maximal number of critical values
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| Title: | On iterates of rational functions with maximal number of critical values |
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| Authors: | Pakovich, Fedor |
| Source: | Journal d'Analyse Mathématique. 156:213-251 |
| Publication Status: | Preprint |
| Publisher Information: | Springer Science and Business Media LLC, 2025. |
| Publication Year: | 2025 |
| Subject Terms: | Mathematics - Complex Variables, FOS: Mathematics, Dynamical Systems (math.DS), Mathematics - Dynamical Systems, 0101 mathematics, Complex Variables (math.CV), 01 natural sciences |
| Description: | Let F be a rational function of one complex variable of degree m ≥ 2. The function F is called simple if for every z ∈ ℂℙ1 the preimage F −1{z} contains at least m − 1 points. We show that if F is a simple rational function of degree m ≥ 4 and F ◦l = G r ◦ G r−1 ◦ ⋯ ◦ G 1, l ≥ 1, is a decomposition of an iterate of F into a composition of indecomposable rational functions, then r = l and there exist Möbius transformations μ i , 1 ≤ i ≤ r − 1, such that G r = F ◦ μ r−1, G i = μ i −1 ◦ F ◦ μ i−1, 1 < i < r, and G 1 = μ 1 −1 ◦ F. As applications, we solve a number of problems in complex and arithmetic dynamics for “general” rational functions. |
| Document Type: | Article |
| Language: | English |
| ISSN: | 1565-8538 0021-7670 |
| DOI: | 10.1007/s11854-025-0386-z |
| DOI: | 10.48550/arxiv.2107.05963 |
| Access URL: | http://arxiv.org/abs/2107.05963 |
| Rights: | CC BY arXiv Non-Exclusive Distribution |
| Accession Number: | edsair.doi.dedup.....f02d29f105878913709c81165160f83f |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Let F be a rational function of one complex variable of degree m ≥ 2. The function F is called simple if for every z ∈ ℂℙ1 the preimage F −1{z} contains at least m − 1 points. We show that if F is a simple rational function of degree m ≥ 4 and F ◦l = G r ◦ G r−1 ◦ ⋯ ◦ G 1, l ≥ 1, is a decomposition of an iterate of F into a composition of indecomposable rational functions, then r = l and there exist Möbius transformations μ i , 1 ≤ i ≤ r − 1, such that G r = F ◦ μ r−1, G i = μ i −1 ◦ F ◦ μ i−1, 1 < i < r, and G 1 = μ 1 −1 ◦ F. As applications, we solve a number of problems in complex and arithmetic dynamics for “general” rational functions. |
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| ISSN: | 15658538 00217670 |
| DOI: | 10.1007/s11854-025-0386-z |