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Semigroups of holomorphic functions; rectifiability and Lipschitz properties of the orbits

Part of: Entire and meromorphic functions, and related topics Real functions

Published online by Cambridge University Press:  28 July 2025

Dimitrios Betsakos  and
Konstantinos Zarvalis
Dimitrios Betsakos
Affiliation:
Department of Mathematics, https://ror.org/02j61yw88 Aristotle University of Thessaloniki, Thessaloniki 54124, Greece e-mail: betsakos@math.auth.gr
Konstantinos Zarvalis*
Affiliation:
Instituto de Matemáticas de la Universidad de Sevilla, Avenida de Reina Mercedes, s/n, Seville 41012, Spain e-mail: kzarvalis@us.es
*
e-mail: zarkonath@math.auth.gr
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Abstract

Let $(\phi _t)$ be a continuous semigroup of holomorphic functions in the unit disk. We prove that all its orbits are rectifiable and that its forward orbits are Lipschitz curves. Moreover, we find a necessary and sufficient condition in terms of hyperbolic geometry so that a backward orbit is a Lipschitz curve. We further explore the Lipschitz condition for forward orbits lying on the unit circle and then for semigroups of holomorphic functions in general simply connected domains.

Keywords

Semigroup of holomorphic functionsLipschitz curveinfinitesimal generatorbackward orbithyperbolic metric

MSC classification

Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Secondary: 26A16: Lipschitz (Hölder) classes

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 30
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

K. Zarvalis is supported by Junta de Andalucía, grant number QUAL21 005 USE.

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