Abstract
Let (ϕt)t≥ 0 be a semigroup of holomorphic functions in the unit disk \(\mathbb {D}\) and K a compact subset of \(\mathbb {D}\). We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk.
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Acknowledgments
The authors would like to thank the anonymous referee for their suggestion to include backward orbits of compact sets under an elliptic semigroup of holomorphic self-maps of the unit disk, as well as, their useful comments.
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Open Access funding enabled and organized by Projekt DEAL. The first author is partially supported by the Alexander von Humboldt Foundation.
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Kourou, M., Zarvalis, K. Compact Sets in Petals and Their Backward Orbits Under Semigroups of Holomorphic Functions. Potential Anal 59, 1913–1939 (2023). https://doi.org/10.1007/s11118-022-10036-7
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DOI: https://doi.org/10.1007/s11118-022-10036-7
Keywords
- Semigroup of holomorphic functions
- Backward orbit
- Petal
- Harmonic measure
- Condenser capacity
- Koenigs function
- Green energy
- Hyperbolic area