Multidimensional conformality and hyperbolic distortion in holomorphic dynamics
Abstract
This manuscript generalizes the concepts of hyperbolic distortion and boundary conformality from the unit disk to multidimensional complex domains such as polydiscs and bounded symmetric domains with intrinsic hyperbolic metrics. We extend strong and weak conformality to higher dimensions, characterize equality cases in multidimensional Schwarz–Pick-type inequalities, and develop invariant distortion metrics. By utilizing multidimensional reproducing kernel Hilbert spaces, we provide operator-theoretic characterizations and investigate applications in holomorphic dynamical systems, including the study of backward orbits and pre-model regularity. These results open pathways to further generalizations in quasiconformal mappings and their multidimensional rigidity properties.
Keywords
References
- Beardon, A. F., Minda, D., The Schwarz–Pick lemma for derivatives, Complex Var. Theory Appl., 49 (2004), 429–454.
- Cowen, C. C., MacCluer, B. D., Composition Operators on Spaces of Analytic Functions, CRC Press, 1995.
- Gumenyuk, P., Kourou, M., Moucha, A., Roth, O., Hyperbolic distortion and conformality at the boundary, Adv. Math., 470 (2025), 110251.
- Gumenyuk, P., Kourou, M., Roth, O., The angular derivative problem for petals of one-parameter semigroups in the unit disk, Rev. Mat. Iberoamericana, 40(3) (2024), 1149-1183.
- Kobayashi, S., Hyperbolic Complex Spaces, Springer-Verlag, 1998.
- Nikolov, N., Pflug, P. Boundary behavior of holomorphic functions, Math. Z., 301 (2022), 567–587.
- Poggi-Corradini, P., Hyperbolic geometry and holomorphic dynamics, Trans. Amer. Math. Soc., 356 (2004), 495–514.
- Shabani, M. M., Hashemi Sababe, S., Coefficient bounds for a subclass of biunivalent functions associated with Dziok-Srivastava operator, Korean J. Math., 30(1) (2022), 73–80.
Details
Primary Language
English
Subjects
Real and Complex Functions (Incl. Several Variables)
Journal Section
Research Article
Authors
Publication Date
December 24, 2025
Submission Date
January 24, 2025
Acceptance Date
July 10, 2025
Published in Issue
Year 2025 Volume: 74 Number: 4
APA
Shabani, M. M., & Hashemi Sababe, S. (2025). Multidimensional conformality and hyperbolic distortion in holomorphic dynamics. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 74(4), 594-607. https://doi.org/10.31801/cfsuasmas.1626138
AMA
1.Shabani MM, Hashemi Sababe S. Multidimensional conformality and hyperbolic distortion in holomorphic dynamics. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2025;74(4):594-607. doi:10.31801/cfsuasmas.1626138
Chicago
Shabani, Mohammad Mehdi, and Saeed Hashemi Sababe. 2025. “Multidimensional Conformality and Hyperbolic Distortion in Holomorphic Dynamics”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 74 (4): 594-607. https://doi.org/10.31801/cfsuasmas.1626138.
EndNote
Shabani MM, Hashemi Sababe S (December 1, 2025) Multidimensional conformality and hyperbolic distortion in holomorphic dynamics. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 74 4 594–607.
IEEE
[1]M. M. Shabani and S. Hashemi Sababe, “Multidimensional conformality and hyperbolic distortion in holomorphic dynamics”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 74, no. 4, pp. 594–607, Dec. 2025, doi: 10.31801/cfsuasmas.1626138.
ISNAD
Shabani, Mohammad Mehdi - Hashemi Sababe, Saeed. “Multidimensional Conformality and Hyperbolic Distortion in Holomorphic Dynamics”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 74/4 (December 1, 2025): 594-607. https://doi.org/10.31801/cfsuasmas.1626138.
JAMA
1.Shabani MM, Hashemi Sababe S. Multidimensional conformality and hyperbolic distortion in holomorphic dynamics. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2025;74:594–607.
MLA
Shabani, Mohammad Mehdi, and Saeed Hashemi Sababe. “Multidimensional Conformality and Hyperbolic Distortion in Holomorphic Dynamics”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 74, no. 4, Dec. 2025, pp. 594-07, doi:10.31801/cfsuasmas.1626138.
Vancouver
1.Mohammad Mehdi Shabani, Saeed Hashemi Sababe. Multidimensional conformality and hyperbolic distortion in holomorphic dynamics. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2025 Dec. 1;74(4):594-607. doi:10.31801/cfsuasmas.1626138
