PreprintCC BY 4.0arXiv
On C2 circle diffeomorphisms with rotation numbers of constant type
Abstract
For every irrational rotation number α of constant type and every q>1, we construct a C2 circle diffeomorphism F arbitrarily close to the rigid rotation Rα in the combined C2 and B22,q topology whose normalized conjugacy is not Lipschitz. We also prove that F''∈ B02,1 implies global C1 conjugacy. Thus the thresholds in the Besov summability and weak Dini scales are q=1 and γ=1/2, respectively. These results complement Herman's Baire category theorem on the residual existence of C2 diffeomorphisms without C1 conjugacy by providing quantitative counterexamples and sufficient conditions for differentiable conjugacy.
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BibTeX
@misc{li2026circle,
title = {{On $C^2$ circle diffeomorphisms with rotation numbers of constant type}},
author = {Junhao Li and Lin Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30750},
eprint = {2610.30750},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30750},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On C^2 circle diffeomorphisms with rotation numbers of constant type
AU - Junhao Li
AU - Lin Wang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30750
UR - https://arxiv.org/abs/2610.30750
AB - For every irrational rotation number α of constant type and every q>1, we construct a C^2 circle diffeomorphism F arbitrarily close to the rigid rotation R_α in the combined C^2 and B^2_2,q topology whose normalized conjugacy is not Lipschitz. We also prove that F''∈ B^0_2,1 implies global C^1 conjugacy. Thus the thresholds in the Besov summability and weak Dini scales are q=1 and γ=1/2, respectively. These results complement Herman's Baire category theorem on the residual existence of C^2 diffeomorphisms without C^1 conjugacy by providing quantitative counterexamples and sufficient conditions for differentiable conjugacy.
ER -