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Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps
Dynamical Systems · Submitted 28 Sep 2026
abstract compiled · 2026-10-02 · no language-model review
The authors' main claim We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $\mathbb{R}^n$, $n\geq 2$: either all orbits are relatively compact, or the iterates converge locally uniformly to a single point of the sphere.
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We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $\mathbb{R}^n$, $n\geq 2$: 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 $\overline{\mathbb{R}^n}$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $\mathbb{R}^n$, $n\geq 2$: either all orbits are relatively compact, or the iterates converge locally uniformly to a single point of the sphere. | result 0.86 | 0.96 | 0.92 | asserted 1.00 |
| s1 | No properness, surjectivity, or boundary extension is assumed. | contribution 0.45 | 0.38 | 0.02 | asserted 0.99 |
| s2 | 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. | method 0.86 | 0.55 | 0.04 | asserted 1.00 |
| s3 | 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. | method 0.98 | 0.89 | 0.03 | asserted 1.00 |
| s4 | As an application, a stable Fatou component of a uniformly quasiregular map of $\overline{\mathbb{R}^n}$ 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. | result 0.93 | 0.85 | 0.10 | asserted 1.00 |
| s5 | Non-injective examples of both escaping types are constructed in every dimension. | result 0.81 | 0.94 | 0.03 | asserted 1.00 |
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Original abstract
We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $\mathbb{R}^n$, $n\geq 2$: 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 $\overline{\mathbb{R}^n}$ 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.
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