On Standard perturbations of the Affine twist map
Abstract
For any t∈ℝ, consider the Affine twist map Afft:𝕋2→𝕋2 given by Afft(x,y)=(x+y mod 1, y+t mod 1). The map Afft clearly possesses an invariant foliation by horizontal curves. If t is rational, each leaf is periodic, and if t is irrational, the orbit of each leaf is dense in the torus. In both cases, the vertical rotation set of the proper lift of Afft to the vertical cylinder is reduced to {t}. Now, for k∈ℝ, define fk,t(x,y):=(x+y+ksin (2πx) mod 1,y+ksin (2πx)+t mod 1). We show that: 1) when t=p/q for integers p and q>0, KAM theory implies the existence of a constant kp/q>0 such that for |k|<kp/q, the vertical rotation set of an adequate lift of fk,p/q is just {p/q}. 2) when t is irrational, for any k≠ 0 the vertical rotation set of the adequate lift of fk,t is a non-degenerate interval which contains t in its interior. In other words, it is not easy to build area-preserving twist maps whose vertical rotation sets are reduced to a single irrational number. From Theorem A of \cite{eujul}, such a map needs to have an invariant foliation by Lipschitz graphs over the horizontal coordinate. In particular, all its iterates must satisfy a twist condition. This is precisely what does not hold for fk,t, for all non-zero values of k.
Related
- Persistence of invariant graphs for twist maps under analytic perturbations
- The Zelevinsky involution, co-standard modules for quantum affine algebras and interval symmetric functions
- Generating the twist subgroup of the level $2$ mapping class group of a non-orientable closed surface
- Bounds on Legendrian Double Twist Knots
- On the length conjecture for twist knots
BibTeX
@misc{zanata2026standard,
title = {{On Standard perturbations of the Affine twist map}},
author = {Salvador Addas Zanata},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30665},
eprint = {2609.30665},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30665},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On Standard perturbations of the Affine twist map
AU - Salvador Addas Zanata
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30665
UR - https://arxiv.org/abs/2609.30665
AB - For any t∈ℝ, consider the Affine twist map Aff_t:𝕋^2→𝕋^2 given by Aff_t(x,y)=(x+y mod 1, y+t mod 1). The map Aff_t clearly possesses an invariant foliation by horizontal curves. If t is rational, each leaf is periodic, and if t is irrational, the orbit of each leaf is dense in the torus. In both cases, the vertical rotation set of the proper lift of Aff_t to the vertical cylinder is reduced to {t}. Now, for k∈ℝ, define f_k,t(x,y):=(x+y+ksin (2πx) mod 1,y+ksin (2πx)+t mod 1). We show that: 1) when t=p/q for integers p and q>0, KAM theory implies the existence of a constant k_p/q>0 such that for |k|<k_p/q, the vertical rotation set of an adequate lift of f_k,p/q is just {p/q}. 2) when t is irrational, for any k≠ 0 the vertical rotation set of the adequate lift of f_k,t is a non-degenerate interval which contains t in its interior. In other words, it is not easy to build area-preserving twist maps whose vertical rotation sets are reduced to a single irrational number. From Theorem A of \cite{eujul}, such a map needs to have an invariant foliation by Lipschitz graphs over the horizontal coordinate. In particular, all its iterates must satisfy a twist condition. This is precisely what does not hold for f_k,t, for all non-zero values of k.
ER -