Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps
Abstract
We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in ℝn, n≥ 2: either all orbits are relatively compact, or the iterates converge locally uniformly to a single point of the sphere. No properness, surjectivity, or boundary extension is assumed. The key tool is a Schwarz-Pick lemma valid in all dimensions: every quasiregular map preserving a bounded measurable conformal structure is non-expanding for a modulus metric built from that structure, and uniformly quasiregular maps preserve such a structure by a theorem of Iwaniec and Martin. Since this metric is not geodesic, we pass to an associated chain metric, which we show is Gromov hyperbolic with boundary the sphere, and apply a theorem of Karlsson. As an application, a stable Fatou component of a uniformly quasiregular map of ℝn̅ that is a uniform domain with uniformly perfect boundary is an attracting basin, a rotation domain, or a parabolic basin with a unique boundary fixed point. Non-injective examples of both escaping types are constructed in every dimension.
Related
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- Examples of Uniformly Quasiregular Mappings with Fatou Set Having Infinitely Many Components
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BibTeX
@misc{fletcher2026schwarzpick,
title = {{Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps}},
author = {Alastair N. Fletcher},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00312},
eprint = {2610.00312},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00312},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps
AU - Alastair N. Fletcher
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00312
UR - https://arxiv.org/abs/2610.00312
AB - We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in ℝ^n, n≥ 2: either all orbits are relatively compact, or the iterates converge locally uniformly to a single point of the sphere. No properness, surjectivity, or boundary extension is assumed. The key tool is a Schwarz-Pick lemma valid in all dimensions: every quasiregular map preserving a bounded measurable conformal structure is non-expanding for a modulus metric built from that structure, and uniformly quasiregular maps preserve such a structure by a theorem of Iwaniec and Martin. Since this metric is not geodesic, we pass to an associated chain metric, which we show is Gromov hyperbolic with boundary the sphere, and apply a theorem of Karlsson. As an application, a stable Fatou component of a uniformly quasiregular map of ℝ^n̅ that is a uniform domain with uniformly perfect boundary is an attracting basin, a rotation domain, or a parabolic basin with a unique boundary fixed point. Non-injective examples of both escaping types are constructed in every dimension.
ER -