TY - JOUR

T1 - Stoilow Factorization of the Heisenberg Group

T2 - Факторизация Стоилова на группе гейзенберга

AU - Dorokhin, D.k.

N1 - Dorokhin, D. K. Stoilow Factorization of the Heisenberg Group / D. K. Dorokhin // Vladikavkaz Mathematical Journal. – 2025. – Vol. 27, No. 3. – P. 50-59. – DOI 10.46698/o8833-7719-4418-f. The work was supported by the Russian Science Foundation, project № 24-21-00319.

PY - 2025/9/18

Y1 - 2025/9/18

N2 - In this article we study the properties of quasiconformal mappings on the Heisenberg group H1 and consider the definition of quasiconformal mappings in terms of the Beltrami equation. In particular, we obtain an explicit expression for the Beltrami coefficient for the composition of two quasiconformal mappings and we prove an analogue of the Stoilow factorization theorem on the plane. Namely, if the Beltrami coefficients of two quasiconformal mappings are equal almost everywhere, then there exists a conformal mapping such that by acting on one of the given quasiconformal mappings from the left, we obtain another given mapping. As an application of these results on the Heisenberg group H1 we compute the Beltrami coefficients of some quasiconformal mappings and we prove a theorem on the images of quasi-Brownian motions. In specific examples we demonstrate the invariance of the Beltrami coefficient under the action of the composition of a conformal function on the corresponding left mapping. Using the Stoilow factorization on the Heisenberg group, we show that if two quasi-Brownian motions have the corresponding Beltrami coefficients equal almost everywhere, then their trajectories are equivalent only if the conformal map in the Stoilov factorization is a map obtained from a composition of translations, rotations and dilations.

AB - In this article we study the properties of quasiconformal mappings on the Heisenberg group H1 and consider the definition of quasiconformal mappings in terms of the Beltrami equation. In particular, we obtain an explicit expression for the Beltrami coefficient for the composition of two quasiconformal mappings and we prove an analogue of the Stoilow factorization theorem on the plane. Namely, if the Beltrami coefficients of two quasiconformal mappings are equal almost everywhere, then there exists a conformal mapping such that by acting on one of the given quasiconformal mappings from the left, we obtain another given mapping. As an application of these results on the Heisenberg group H1 we compute the Beltrami coefficients of some quasiconformal mappings and we prove a theorem on the images of quasi-Brownian motions. In specific examples we demonstrate the invariance of the Beltrami coefficient under the action of the composition of a conformal function on the corresponding left mapping. Using the Stoilow factorization on the Heisenberg group, we show that if two quasi-Brownian motions have the corresponding Beltrami coefficients equal almost everywhere, then their trajectories are equivalent only if the conformal map in the Stoilov factorization is a map obtained from a composition of translations, rotations and dilations.

KW - Группа Гейзенберга

KW - Факторизация Стоилова

KW - Квазиконформные отображения

KW - Система Бельтрами

KW - Броуновское движение

KW - HEISENBERG GROUP

KW - STOILOW FACTORIZATION

KW - quasiconformal mappings

KW - BELTRAMI SYSTEM

UR - https://elibrary.ru/item.asp?id=82937289

U2 - 10.46698/o8833-7719-4418-f

DO - 10.46698/o8833-7719-4418-f

M3 - Article

VL - 27

SP - 50

EP - 59

JO - Владикавказский математический журнал

JF - Владикавказский математический журнал

SN - 1683-3414

IS - 3

ER -