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Dynamics of Hyperbolic Schwarz Reflections
Abstract
We give a conformal mating description for the class Σd of hyperbolic Schwarz reflections with connected and full filled Julia set; this parallels the recent development in several non-hyperbolic settings. We show that the escaping dynamics of Σd can be characterized independently by the class of \emph{annular Schwarz reflections} we introduce. Our main result is that each σ∈Σd decomposes into a unique hyperbolic polynomial factor and a unique annular Schwarz reflection factor and, conversely, any two such factors may be mated to determine a unique such Schwarz reflection. We also show that annular Schwarz reflections converge to Nielsen maps in suitable settings, connecting the hyperbolic and non-hyperbolic mating descriptions.
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BibTeX
@misc{lazebnik2026dynamics,
title = {{Dynamics of Hyperbolic Schwarz Reflections}},
author = {Kirill Lazebnik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40196},
eprint = {2609.40196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40196},
}
RIS
TY - UNPB
TI - Dynamics of Hyperbolic Schwarz Reflections
AU - Kirill Lazebnik
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40196
UR - https://arxiv.org/abs/2609.40196
AB - We give a conformal mating description for the class Σ_d of hyperbolic Schwarz reflections with connected and full filled Julia set; this parallels the recent development in several non-hyperbolic settings. We show that the escaping dynamics of Σ_d can be characterized independently by the class of \emph{annular Schwarz reflections} we introduce. Our main result is that each σ∈Σ_d decomposes into a unique hyperbolic polynomial factor and a unique annular Schwarz reflection factor and, conversely, any two such factors may be mated to determine a unique such Schwarz reflection. We also show that annular Schwarz reflections converge to Nielsen maps in suitable settings, connecting the hyperbolic and non-hyperbolic mating descriptions.
ER -