Geometric Orbital Linearization of Siegel Singularities in the Plane
Abstract
We give geometric characterizations of analytically linearizable nondegenerate Siegel foliations in C2 using complex tangencies with shrinking strictly pseudoconvex hypersurfaces. Our principal result treats a fixed bounded smoothly bounded strictly pseudoconvex Reinhardt domain: in coordinates compatible with the eigendirections of the linear part, two-dimensional tangency loci along a shrinking sequence characterize analytic Siegel linearizability. For round spheres we prove a stronger statement: no a priori alignment of the Euclidean coordinate axes with the eigendirections is required; the tangency hypothesis itself forces the linear part to be unitarily diagonalizable with negative real eigenvalue ratio. We also establish a polynomial Shilov-boundary criterion on a round sphere and a logarithmic one-boundary criterion valid for arbitrary smooth strictly pseudoconvex domains, provided the embedded closed bidisc lies on the pseudoconvex side. Finally, we study the transverse-holomorphic dynamics of smooth tangency tori, including a rigidity theorem in the periodic case.
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BibTeX
@misc{scardua2026geometric,
title = {{Geometric Orbital Linearization of Siegel Singularities in the Plane}},
author = {Toshikazu Ito \and Bruno Scárdua},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14855},
eprint = {2609.14855},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14855},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric Orbital Linearization of Siegel Singularities in the Plane
AU - Toshikazu Ito \and Bruno Scárdua
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14855
UR - https://arxiv.org/abs/2609.14855
AB - We give geometric characterizations of analytically linearizable nondegenerate Siegel foliations in C^2 using complex tangencies with shrinking strictly pseudoconvex hypersurfaces. Our principal result treats a fixed bounded smoothly bounded strictly pseudoconvex Reinhardt domain: in coordinates compatible with the eigendirections of the linear part, two-dimensional tangency loci along a shrinking sequence characterize analytic Siegel linearizability. For round spheres we prove a stronger statement: no a priori alignment of the Euclidean coordinate axes with the eigendirections is required; the tangency hypothesis itself forces the linear part to be unitarily diagonalizable with negative real eigenvalue ratio. We also establish a polynomial Shilov-boundary criterion on a round sphere and a logarithmic one-boundary criterion valid for arbitrary smooth strictly pseudoconvex domains, provided the embedded closed bidisc lies on the pseudoconvex side. Finally, we study the transverse-holomorphic dynamics of smooth tangency tori, including a rigidity theorem in the periodic case.
ER -