Weak Asymptotic Symmetry of Quasisymmetric Embeddings on Quasilines
Abstract
We investigate the relationship between weak asymptotic symmetry (WAS) and asymptotic symmetry (AS) for quasisymmetric maps associated with planar quasilines. To this end, we introduce a formally weaker condition, called equidistant weak asymptotic symmetry (EWAS), and prove that, for quasisymmetric embeddings of ℝ into ℂ, the three conditions AS, WAS, and EWAS are equivalent. We then establish that WAS implies AS for quasisymmetric homeomorphisms from an arbitrary quasiline onto ℝ. More generally, if h:Γ1→Γ2 is a quasisymmetric homeomorphism between quasilines and Γ1 is asymptotically conformal, then WAS implies AS. A formally dual statement holds when Γ2 is asymptotically conformal, provided that h is uniformly continuous. In the compact setting of bounded quasicircles, these results yield an answer to the WAS-AS problem posed by Brania and Yang.
Related
- Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence
- Asymptotic observable symmetry for spacetime observability of waves II: Smooth bounded domains
- Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers
- Asymptotic observable symmetry for spacetime observability of waves I: The disc
- a-contraction applied to the time-asymptotic stability of weak shocks for a general 1-D viscous system
BibTeX
@misc{matsuzaki2026weak,
title = {{Weak Asymptotic Symmetry of Quasisymmetric Embeddings on Quasilines}},
author = {Katsuhiko Matsuzaki and Fei Tao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19733},
eprint = {2609.19733},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19733},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weak Asymptotic Symmetry of Quasisymmetric Embeddings on Quasilines
AU - Katsuhiko Matsuzaki
AU - Fei Tao
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19733
UR - https://arxiv.org/abs/2609.19733
AB - We investigate the relationship between weak asymptotic symmetry (WAS) and asymptotic symmetry (AS) for quasisymmetric maps associated with planar quasilines. To this end, we introduce a formally weaker condition, called equidistant weak asymptotic symmetry (EWAS), and prove that, for quasisymmetric embeddings of ℝ into ℂ, the three conditions AS, WAS, and EWAS are equivalent. We then establish that WAS implies AS for quasisymmetric homeomorphisms from an arbitrary quasiline onto ℝ. More generally, if h:Γ_1→Γ_2 is a quasisymmetric homeomorphism between quasilines and Γ_1 is asymptotically conformal, then WAS implies AS. A formally dual statement holds when Γ_2 is asymptotically conformal, provided that h is uniformly continuous. In the compact setting of bounded quasicircles, these results yield an answer to the WAS-AS problem posed by Brania and Yang.
ER -