Families of invariant tori in KAM theory: interplay of integer characteristics (Q1702805)

The paper provides a short but very dense review of several aspects of KAM theory (for which the author provided several deep results) in different contexts, such as Hamiltonian isotropic, volume-preserving, dissipative and reversible. Special attention is give to the so-called ``reversible 2 context''. In the reversible case, there is an involution \(G: M \to M\) and a \(G\)-reversible vector field \(V\) on \(M\); call \(a\) the dimension of the fixed point set \(F(G)\) for \(G\) and assume all its connected components have the same dimension. If a quasi-periodic invariant \(n\)-torus \(\mathcal{T}\) of \(V\) exists and is also \(G\)-invariant, then \(F(G) \cap \mathcal{T}\) is made of isolated points, and the codimension of any component of \(F(G)\) intersecting \(\mathcal{T}\) is at least \(n\), call it \(n+b\). Then the {``reversible context 1''} is the case with \(a \geq b\), while the {``reversible context 2''} is that with \(a < b\). Fundamental results for this case have been obtained only recently [the author, Regul. Chaotic Dyn. 21, No. 6, 599--620 (2016; Zbl 1369.70044)], and are put in perspective in the present paper.