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

Published online by Cambridge University Press:  28 July 2025

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

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.

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Type
Article
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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