Around the stability of KAM tori (Q2354907)

Let \[ H(\varphi,r)=\langle\omega_0,r\rangle+\mathcal{O}(r^2)\tag{1.1} \] be a \(C^2\) function defined for \(( \varphi,r)\in \mathbb{T}^d \times \mathbb{R}^d\) and \(r\sim 0 \). The Hamiltonian system associated to \(H\) is given by \[ \begin{cases}\dot{\varphi}= \partial_r H(\varphi,r), \\ \dot{r}= -\partial_\varphi H(\varphi,r). \end{cases}\tag{1.2} \] System (1.2) is integrable and admits invariant tori \(\mathcal{T}_0=\mathbb{T}^d \times \{0\}\) with the frequency vector \(\omega_0 \) that satisfies the Diophantine condition: \[ |\langle k, \omega_0 \rangle|\geq \frac{\kappa}{|k|^\tau} , \quad \forall k\in \mathbb{Z}^n \backslash \{0\}, \tag{1.3} \] where \(\kappa>0\) and \(\tau>d-1\) are some constants. By definition, a \(C^r\) (resp. smooth or analytic) invariant torus with an induced flow that is \(C^r\) (resp. smoothly or analytically) conjugated to a Diophantine translation is a \(C^r\) (resp. smooth or analytic) KAM torus of (1.2) with translation vector \(\omega\). \(\mathcal{T}_0\) is KAM stable in the strong sense if \(\mathcal{T}_0\) is accumulated by invariant tori whose Lebesgue density in the phase space tends to one in the neighborhood of \(\mathcal{T}_0\) and whose frequencies cover a set of positive measure. \(\mathcal{T}_0\) is KAM stable in a weaker sense if we drop the requirement that the frequencies cover a set of positive measure. The KAM stability of the torus \(\mathcal{T}_0\) is studied under different hypotheses on \(H\) and \(\omega_0\) and the following results are obtained in this work: {\parindent=0.8cm \begin{itemize}\item[(i)] If \(\omega_0\) is Diophantine and \(H\) is analytic, then \(\mathcal{T}_0\) is accumulated by KAM tori. \item[(ii)] If \(\omega_0\) is Diophantine, \(H\) is analytic and the Birkhoff normal form of \(H\) satisfies a Rüssmann transversality condition at \(\mathcal{T}_0\), then \(\mathcal{T}_0\) is KAM stable in the weaker sense. \item[(iii)] In two degrees of freedom (\(d = 2\)), if the components of the frequency vector \(\omega_0\) are rationally independent and if \(H\) satisfies a Kolmogorov nondegeneracy condition of its Hessian matrix at \(\mathcal{T}_0\), then \(\mathcal{T}_0\) is KAM stable. For \(d\geq 3\) one has KAM stability for a class of \(\mathcal{T}_0\) that includes all vectors except a meagre set of zero Hausdorff dimension. \item[(iv)] For \(d\geq 4\), for any \(\omega_0\in \mathbb{R}^d\) , there exists a \(C^\infty\) (Gevrey) \(H \) as in (1.2), such that \(\mathcal{T}_0\) is not KAM stable (no positive measure of accumulating tori). \item[(v)] For \(d = 2\), if \(\omega_0\) is Diophantine and \(H\) is smooth, then \(\mathcal{T}_0\) is KAM stable. \end{itemize}} The proofs are based on a counterterm KAM theorem formulated and discussed in the paper.