Jet rigidity of Mather's β-function and complexified KAM curves holomorphic in potential for analytic standard maps
Abstract
Mather's β-function associates with each rotation number the least average action carried by an orbit of a twist map. For analytic standard maps in the KAM regime, we prove the following. For any family of d ≥ 1 directions of perturbation satisfying a natural symmetry condition, the jet of the β-function at a single algebraic Diophantine rotation number locally determines the potential for an open and prevalent, hence dense, set of base potentials in the KAM domain. In particular, we obtain that for a residual and prevalent set of even potentials in the KAM domain, every real analytic deformation with finite Fourier support that preserves the full jet of the β-function at a fixed algebraic Diophantine rotation number is constant. These results are non perturbative within the KAM domain. As an intermediate result of independent interest, we establish a KAM theorem giving joint C∞--holomorphic dependence of the invariant curves, and hence of the β-function, on complex holomorphic potentials and on a domain of complexified rotation numbers whose boundary contains real Diophantine numbers.
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BibTeX
@misc{helfter2026rigidity,
title = {{Jet rigidity of Mather's $β$-function and complexified KAM curves holomorphic in potential for analytic standard maps}},
author = {Mathieu Helfter},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15617},
eprint = {2609.15617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15617},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Jet rigidity of Mather's β-function and complexified KAM curves holomorphic in potential for analytic standard maps
AU - Mathieu Helfter
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15617
UR - https://arxiv.org/abs/2609.15617
AB - Mather's β-function associates with each rotation number the least average action carried by an orbit of a twist map. For analytic standard maps in the KAM regime, we prove the following. For any family of d ≥ 1 directions of perturbation satisfying a natural symmetry condition, the jet of the β-function at a single algebraic Diophantine rotation number locally determines the potential for an open and prevalent, hence dense, set of base potentials in the KAM domain. In particular, we obtain that for a residual and prevalent set of even potentials in the KAM domain, every real analytic deformation with finite Fourier support that preserves the full jet of the β-function at a fixed algebraic Diophantine rotation number is constant. These results are non perturbative within the KAM domain. As an intermediate result of independent interest, we establish a KAM theorem giving joint C^∞--holomorphic dependence of the invariant curves, and hence of the β-function, on complex holomorphic potentials and on a domain of complexified rotation numbers whose boundary contains real Diophantine numbers.
ER -