The chain recurrent set of flow of automorphisms on a decomposable Lie group

TL;DR

The paper investigates the chain recurrent set $\mathcal{R}_C(\varphi)$ for flows of automorphisms on connected Lie groups, proving that if the group is decomposable by the flow, then $\mathcal{R}_C(\varphi)=G^0$, the central subgroup fixed by the hyperbolic part. It leverages the Jordan decomposition at both the algebra and group levels, uniform neighborhoods, and projections to homogeneous spaces to reduce to nilpotent flows and analyze their recurrence properties. The authors establish the restriction property $\mathcal{R}_C(\varphi)=\mathcal{R}_C(\varphi|_{\mathcal{R}_C(\varphi)})$ and prove chain transitivity for nilpotent flows on solvable, compact semisimple, and noncompact semisimple groups, culminating in a general result for decomposable $G$. These results provide a precise classification of recurrence in flows of automorphisms on decomposable Lie groups and offer a framework for understanding chain components via the hyperbolic, elliptic, and nilpotent parts of the flow.

Abstract

In this paper we show that the chain recurrent set of a flow of automorphisms on a connected Lie group coincides with the central subgroup of the flow, if the group is decomposable. Moreover, in the decomposable case, the flow satisfies the restriction property. Furthermore, the restriction of any flow of automorphisms to the connected component of the identity of its central subgroup is chain transitive.