Abstract
It has been an open problem for about 10 years whether every trajectory of a parabolic one-parameter semigroup in the unit disk tends to the Denjoy–Wolff point with a definite (and common for all trajectories) slope. In this paper, we give the negative answer to this question.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Notice that for any parabolic one-parameter semigroup, \(\angle \lim _{z\rightarrow \tau } \frac{G(z)}{z-\tau } =0\).
References
Abate, M.: Iteration theory of holomorphic maps on taut manifolds. In: Research and Lecture Notes in Mathematics. Complex Analysis and Geometry, Mediterranean, Rende. MR1098711 (92i:32032) (1989)
Abate, M., Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: The evolution of Loewner’s differential equations. Newslett. Eur. Math. Soc. 78, 31–38 (2010)
Berkson, E., Porta, H.: Semigroups of analytic functions and composition operators. Michigan Math. J. 25(1), 101–115 (1978). MR0480965 (58 #1112)
Contreras, M.D., Díaz-Madrigal, S.: Analytic flows in the unit disk: angular derivatives and boundary fixed points. Pac. J. Math. 222, 253–286 (2005)
Contreras, M.D., Díaz-Madrigal, S., Pommerenke, Ch.: Some remarks on the Abel equation in the unit disk. J. Lond. Math. Soc. (2) 75, 623–634 (2007)
Conway, J.B.: Functions of One Complex Variable, vol. II, 2nd edn (Graduate Texts in Mathematics, vol. 159). Springer, Berlin (1996)
Elin, M., Jacobson, F.: Parabolic type semigroups: asymptotics and order of contact. Available at http://arxiv.org/pdf/1309.4002.pdf (2013, preprint)
Elin, M., Khavinson, D., Reich, S., Shoikhet, D.: Linearization models for parabolic dynamical systems via Abel’s functional equation. Ann. Acad. Sci. Fen. 35, 1–34 (2010)
Elin, M., Reich, S., Shoikhet, D., Yacobson, F.: Asymptotic behavior of one-parameter semigroups and rigidity of holomorphic generators. Complex Anal. Oper. Theory 2, 55–86 (2008)
Elin, M., Shoikhet, D.: Linearization models for complex dynamical systems. Topics in univalent functions, functional equations and semigroup theory. In: Operator Theory: Advances and Applications, vol. 208. Birkhäuser, Basel (2010)
Elin, M., Shoikhet, D.: Boundary behavior and rigidity of semigroups of holomorphic mappings. Anal. Math. Phys. 1, 241–258 (2011)
Elin, M., Shoikhet, D., Yacobson, F.: Linearization models for parabolic type semigroups. J. Nonlinear Convex Anal. 9, 205–214 (2008)
Goryaĭnov, V.V.: Semigroups of analytic functions in analysis and applications. Uspekhi Mat. Nauk 67(6(408)), 5–52 (2012). (translation in Russian Math. Surv. 67(6), 975–1021 (2012) MR3075076)
Poggi-Corradini, P.: Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk. Rev. Mat. Iberoam. 19, 943–970 (2003)
Pommerenke, Ch.: On the iteration of analytic functions in a half plane, I. J. Lond. Math. Soc. (2) 19, 439–447 (1979)
Shapiro, J.H.: Composition operators and classical function theory. In: Universitext: Tracts in Mathematics. Springer, New York (1993)
Shoikhet, D.: Semigroups in Geometrical Function Theory. Kluwer Academic Publishers, Dordrecht (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Lawrence Zalcman.
M.D. Contreras and S. Díaz-Madrigal were partially supported by the Ministerio de Economía y Competitividad and the European Union (FEDER), project MTM2012-37436-C02-01, and by La Consejería de Educación y Ciencia de la Junta de Andalucía. P. Gumenyuk was partially supported by the FIRB grant Futuro in Ricerca “Geometria Differenziale Complessa e Dinamica Olomorfa” n. RBFR08B2HY.
Rights and permissions
About this article
Cite this article
Contreras, M.D., Díaz-Madrigal, S. & Gumenyuk, P. Slope Problem for Trajectories of Holomorphic Semigroups in the Unit Disk. Comput. Methods Funct. Theory 15, 117–124 (2015). https://doi.org/10.1007/s40315-014-0092-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40315-014-0092-9