Sobolev mixing constants and compactness in rational moduli space
Abstract
We study uniformity of Sobolev mixing estimates for rational maps and uniformly quasiregular mappings. For rational maps of fixed degree, the optimal centered Sobolev trace constant Af is a continuous proper function on Möbius moduli space. Uniform bounds on Af therefore characterize relative compactness in moduli, and a minimizing class exists in every degree. In the setting of uniformly quasiregular endomorphisms of degree d≥2 on closed n-manifolds with an invariant conformal structure, the kth centered transfer operator from critical Sobolev energy into L1 of the equilibrium measure has norm Af d−k/n. On the mean-zero Sobolev space, the spectrum and Fredholm essential spectrum are the closed disk of radius d−1/n, with infinite-dimensional eigenspaces throughout its interior. The proofs use the energy scaling of pullback, a bounded equilibrium trace, and an obstruction from atoms of intermediate mass. Combined with conformal barycenter normalization and DeMarco--Faber's degeneration theorem, this obstruction gives the moduli compactness criterion. Explicit families illustrate the distinction between degeneration and concentration caused by changes of coordinates.
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BibTeX
@misc{fletcher2026sobolev,
title = {{Sobolev mixing constants and compactness in rational moduli space}},
author = {Alastair Fletcher and Ilya Krishtal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19301},
eprint = {2609.19301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19301},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sobolev mixing constants and compactness in rational moduli space
AU - Alastair Fletcher
AU - Ilya Krishtal
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19301
UR - https://arxiv.org/abs/2609.19301
AB - We study uniformity of Sobolev mixing estimates for rational maps and uniformly quasiregular mappings. For rational maps of fixed degree, the optimal centered Sobolev trace constant A_f is a continuous proper function on Möbius moduli space. Uniform bounds on A_f therefore characterize relative compactness in moduli, and a minimizing class exists in every degree. In the setting of uniformly quasiregular endomorphisms of degree d≥2 on closed n-manifolds with an invariant conformal structure, the kth centered transfer operator from critical Sobolev energy into L^1 of the equilibrium measure has norm A_f d^−k/n. On the mean-zero Sobolev space, the spectrum and Fredholm essential spectrum are the closed disk of radius d^−1/n, with infinite-dimensional eigenspaces throughout its interior. The proofs use the energy scaling of pullback, a bounded equilibrium trace, and an obstruction from atoms of intermediate mass. Combined with conformal barycenter normalization and DeMarco--Faber's degeneration theorem, this obstruction gives the moduli compactness criterion. Explicit families illustrate the distinction between degeneration and concentration caused by changes of coordinates.
ER -