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arXiv · 2501.01952

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

Dimitrios Betsakos·Konstantinos Zarvalis
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Abstract

Let $(\phi_t)$ be a 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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Dimitrios Betsakos, Konstantinos Zarvalis. 2025-01-03. Semigroups of holomorphic functions; rectifiability and Lipschitz properties of the orbits. https://doi.org/10.4153/s0008414x25101223

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