Resumen de Invariant Tori for Area-Preserving Maps with Ultra-differentiable Perturbation and Liouvillean Frequency

Hongyu Cheng, Fenfen Wang, Shimin Wang

• We prove the existence of invariant tori to the area-preserving maps defined on R2 ×T x = F(x, θ ), θ = θ + α (α ∈ R\Q), where F is related to a linear rotation, and the perturbation is ultra-differentiable in θ ∈ T, which is very closed to C∞ regularity. Moreover, we assume that the frequency α is any irrational number without other arithmetic conditions and the smallness of the perturbation does not depend on α. Thus, both the difficulties from the ultra-differentiability of the perturbation and Liouvillean frequency will appear in this work. The proof of the main result is based on the Kolmogorov-Arnold-Moser (KAM) scheme about the area-preserving maps with some new techniques.