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S. A. Zanata

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Preprint Sep 2026

On Standard perturbations of the Affine twist map

For any $t\in\mathbb{R}$, consider the Affine twist map $\rm{Aff_t}:\mathbb{T}^2\to\mathbb{T}^2$ given by $$\rm{Aff_t}(x,y)=(x+y \text{ mod 1}, y+t \text{ mod 1}).$$ The map $\rm{Aff_t}$ clearly possesses an invariant foliation by horizontal curves. If $t$ is rational, each leaf is periodic, and if $t$ is irrational, t...

S. A. Zanata · 0 citations

Preprint Sep 2026

Foliations and minimality for twist maps

In this paper we consider area-preserving twist maps $f:\mathbf{T}^2\to\mathbf{T}^2$ with vertical rotation sets reduced to a single irrational number $\alpha$. Our first result states that under these hypotheses, there exists a $f$-invariant foliation $\mathcal{F}_G=\{\operatorname{Graph}(\phi_t)\}_{t\in\mathbf{T}^1}$...

S. A. Zanata, Julian Lazaro Aguirre · 1 citation