Periodic orbits of Linear and Invariant flows on Semisimple Lie groups

S. N. Stelmastchuk

Published 2020-02-19Version 1

Our main is to study periodic orbits of linear or invariant flows on a real, connected, semisimple Lie group. Since there exist a derivation of Lie algebra to linear or invariant flow, we show that a periodic orbit that is not fixed point of a linear or invariant flow is periodic if and only the eingevalues of derivation is 0 or $\pm \alpha i$ for an unique $\alpha \neq 0$ and they are semisimple. We apply this result in noncompact case through Iwasawa's decomposition. Furthermore, we present a version of Poincar\'e-Bendixon's Theorem for periodic orbits.