Recurrent orbits of subgroups of local complex analytic diffeomorphisms

TL;DR

This paper demonstrates that non-virtually solvable subgroups of local complex analytic diffeomorphisms always have recurrent orbits, revealing a fundamental dynamical property of such groups in complex analysis.

Contribution

It establishes the recurrence of orbits for non-virtually solvable subgroups of local complex analytic diffeomorphisms, a new insight in complex dynamical systems.

Findings

Non-virtually solvable subgroups have recurrent orbits

Existence of orbits contained in their set of limit points

Recurrent phenomena linked to subgroup structure

Abstract

We show recurrent phenomena for orbits of groups of local complex analytic diffeomorphisms that have a certain subgroup or image by a morphism of groups that is non-virtually solvable. In particular we prove that a non-virtually solvable subgroup of local biholomorphisms has always recurrent orbits, i.e. there exists an orbit contained in its set of limit points.