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Semigroups of holomorphic functions; rectifiability and Lipschitz properties of the orbits
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 28 July 2025, pp. 1-30
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- Article
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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.
