Discrete orbits, recurrence and solvable subgroups of Diff(C^2,0)

TL;DR

This paper investigates the local dynamics of subgroups of Diff(C^2,0) with discrete orbits, establishing that such groups are virtually solvable, which has implications for integrable systems.

Contribution

It proves that subgroups of Diff(C^2,0) with locally discrete orbits are virtually solvable, advancing understanding of their structure and dynamics.

Findings

Subgroups with locally discrete orbits are virtually solvable.

The structure of the recurrent set is characterized for general groups.

Results have applications in integrable systems.

Abstract

We discuss the local dynamics of a subgroup of Diff(C^2,0) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of Diff(C^2,0) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems.