Discrete orbits, recurrence and solvable subgroups of Diff \((\mathbb {C}^2, 0)\) (Q522463)

The authors discuss the local dynamics of subgroups of the group holomorphic diffeomorphisms fixing the origin, denoted by \(\text{Diff}(\mathbb{C}^2, 0)\), possessing locally discrete orbits as well as the structure of the recurrent sets for more general groups. A group is said to be virtually solvable if it contains a normal, solvable subgroup of finite index. Consider finitely many local holomorphic diffeomorphisms \(f_1, \ldots , f_k\) inducing elements of \(\text{Diff} (\mathbb{C}^2 , 0)\). Denote by \(G\) or simply \(G_U\) the pseudogroup generated by them on some chosen small neighborhood \(U\) of the origin in \(\mathbb{C}^2\). \(G\) is said to have locally discrete orbits (resp., finite orbits), if there is a sufficiently small neighborhood \(U\) of the origin where \(G_U\) has locally discrete orbits (resp., finite orbits). The first main result reads: Theorem A. Suppose that \(G\) is a finitely generated subgroup of \(\text{Diff}(\mathbb{C}^2 , 0)\) with locally discrete orbits. Then \(G\) is virtually solvable. A point \(p \in U\) is said to be recurrent if there exists a sequence \( \{g_n \} \) of elements in \(G_U\) such that \(g_n (p)\to p\) with \(g_n (p) \neq p\) for every \(n\). Consider the normal subgroup \(\text{Diff}_1 (\mathbb{C}^2 , 0)\) consisting of those local holomorphic diffeomorphisms tangent to the identity. When \(G\) happens to be a (pseudo-)subgroup of \(\text{Diff}_1(\mathbb{C}^2, 0)\), the following stronger result holds: Theorem B. Consider a non-solvable group \(G \subset \text{Diff}_1 (\mathbb{C}^2 , 0)\) and denote by \((G)\) the set of points that fail to be recurrent for \(G\). Then there is a neighborhood \(U\) of the origin in \( \mathbb{C}^2\) such that \((G) \cap U\) is contained in a countable union of proper analytic subsets of \(U\) (in particular, \( (G) \cap U\) has null Lebesgue measure). These results are of considerable interest in problems related to integrable systems, [\textit{J. J. Morales-Ruiz} et al., Ann. Sci. Éc. Norm. Supér. (4) 40, No. 6, 845--884 (2007; Zbl 1144.37023)]. In this sense, Theorem A says that, the integrable character of a subgroup of \(\text{Diff}(\mathbb{C}^2 , 0)\) possessing locally finite orbits lies in the fact that this group must be virtually solvable. Relations with other interesting subjects are provided: Shcherbakov-Nakai theory, the analytic limit set problem, and Kleinian groups in an accurate sense. Several interesting examples are described.