A mechanical counterexample to KAM theory with low regularity

KAM, $α$-Gevrey regularity and the $α$-Bruno-Rüssmann condition

We prove a new invariant torus theorem, for $α$-Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the $α$-Bruno-Rüssmann condition, and which reduces to the classical Bruno-Rüssmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this …

走向精确正则性：$ \mathbb{T}^\infty $ 上 $ C^\infty $ 向量场中的全维环面

Towards sharp regularity: Full-dimensional tori in $ C^\infty $ vector fields over $ \mathbb{T}^\infty $

研究无穷维环面上受扰向量场的线性化问题，提出带平衡序列的新 KAM 格式克服非多项式非共振，确定 KAM 适用所需的最弱正则性为 $ C^\infty $ 正则性。

Integrability of moduli and regularity of Denjoy counterexamples

We study the regularity of exceptional actions of groups by $C^{1,α}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $α$. Let $G$ be a finitely generated group admitting a $C^{1,α}$ action $ρ$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/α^d$ is integrable near zero, then under som…

Some counterexamples to the central limit theorem for random rotations

Fix an irrational number $α$, and consider a random walk on the circle in which at each step one moves to $x+α$ or $x-α$ with probabilities $1/2, 1/2$ provided the current position is $x$. If an observable is given we can study a process called an additive functional of this random walk. One can formulate certain relations between the regularity of the observable and the Diophantine properties of $α$ implying the central limit theorem. It is proven here that for every Liouville angle there exists a smooth observable such that the central limit theorem fails. We construct also a Liouville angle…

An a posteriori KAM theorem for whiskered tori in Hamiltonian partial differential equations with applications to some ill-posed equations

The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has an \emph{a-posteriori} format, i.e., we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). The method of proof is based on an iterative method to solve a functiona…

关于强叶列缅科猜想的慢增长反例

Slow-growing counterexamples to the strong Eremenko conjecture

1989年叶列缅科提出强叶列缅科猜想，2011年被反例证伪。本文通过改编通道，推导出满足某些增长性质的叶列缅科-柳比歇夫类\(\mathcal{B}\)反例函数\(f\)的存在性，考虑了其增长能有多慢，给出特定条件下\(f\)增长的结论。