TY - UNPB

T1 - On iterates of rational functions with maximal number of critical values

AU - Pakovich, Fedor

PY - 2021/7/13

Y1 - 2021/7/13

N2 - Let F be a rational function of one complex variable of degree m ≥ 2. The function F is called simple if for each z ∈ CP1 the preimage P−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 = Gr ◦Gr−1 ◦· · · ◦G1, l ≥ 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 µi, 1 ≤ i ≤ r − 1, such that Gr = F ◦µr−1, Gi = µi−1◦F ◦µi−1, 1 < i < r, and G1 = µ−11◦F. As an application, we provide explicit solutions of a number of problems in complex and arithmetic dynamics for “general” rational functions.

AB - Let F be a rational function of one complex variable of degree m ≥ 2. The function F is called simple if for each z ∈ CP1 the preimage P−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 = Gr ◦Gr−1 ◦· · · ◦G1, l ≥ 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 µi, 1 ≤ i ≤ r − 1, such that Gr = F ◦µr−1, Gi = µi−1◦F ◦µi−1, 1 < i < r, and G1 = µ−11◦F. As an application, we provide explicit solutions of a number of problems in complex and arithmetic dynamics for “general” rational functions.

KW - math.DS

KW - math.CV

U2 - 10.48550/arXiv.2107.05963

DO - 10.48550/arXiv.2107.05963

M3 - Preprint

BT - On iterates of rational functions with maximal number of critical values

ER -