TY - UNPB

T1 - On algebraic dependencies between Poincaré functions

AU - Pakovich, Fedor

PY - 2021/6/10

Y1 - 2021/6/10

N2 - Let A be a rational function of one complex variable, and z0 its repelling fixed point with the multiplier λ. Then a Poincaré function associated with z0 is a function PA,z0,λ meromorphic on C such that PA,z0,λ(0)=z0, P′A,z0,λ(0)≠0, and PA,z0,λ(λz)=A∘PA,z0,λ(z). In this paper, we investigate the following problem: given Poincaré functions PA1,z1,λ1 and PA2,z2,λ2, find out if there is an algebraic relation f(PA1,z1,λ1,PA2,z2,λ2)=0 between them and, if such a relation exists, describe the corresponding algebraic curve. We provide a solution, which can be viewed as a refinement of the classical Ritt theorem about commuting rational functions. We also complement previous results concerning algebraic dependencies between Böttcher functions.

AB - Let A be a rational function of one complex variable, and z0 its repelling fixed point with the multiplier λ. Then a Poincaré function associated with z0 is a function PA,z0,λ meromorphic on C such that PA,z0,λ(0)=z0, P′A,z0,λ(0)≠0, and PA,z0,λ(λz)=A∘PA,z0,λ(z). In this paper, we investigate the following problem: given Poincaré functions PA1,z1,λ1 and PA2,z2,λ2, find out if there is an algebraic relation f(PA1,z1,λ1,PA2,z2,λ2)=0 between them and, if such a relation exists, describe the corresponding algebraic curve. We provide a solution, which can be viewed as a refinement of the classical Ritt theorem about commuting rational functions. We also complement previous results concerning algebraic dependencies between Böttcher functions.

KW - math.DS

KW - math.CV

U2 - https://doi.org/10.48550/arXiv.2106.05770

DO - https://doi.org/10.48550/arXiv.2106.05770

M3 - Preprint

BT - On algebraic dependencies between Poincaré functions

ER -