
Journal of Lie Theory 30 (2020), No. 2, 587616 Copyright Heldermann Verlag 2020 Semigroups and Moment Lyapunov Exponents Luiz A. B. San Martin Universidade Estadual de Campinas, Campinas  SP, Brazil smartin@ime.unicamp.br [Abstractpdf] Let $G$ be a noncompact semisimple Lie group with finite center and $\mu $ a probability measure on $G$. We consider (i) the semigroup $S_{\mu }$ generated by the support of $\mu $ (with the assumption that $\mathrm{int}% S_{\mu }\neq \emptyset $); (ii) The spectral radii $r_{\lambda }$ of the operators $U_{\lambda }\left( \mu \right) $ where $U_{\lambda }$ is a (nonunitary) representation of $G$ induced by a real character and (iii) the moment Lyapunov exponents $\gamma \left( \lambda ,x\right) $ of the i.i.d.\ random product on $G$ defined by $\mu $. The equality $r_{\lambda }=\gamma \left( \lambda ,x\right) $ holds in many cases. We give a necessary and sufficient condition to have $S_{\mu }=G$ in terms of the analyticity of the map $\lambda \mapsto r_{\lambda }$. The condition is applied to measures obtained by solutions of invariant stochastic differential equations on $G$ yielding a necessary and sufficient condition for the controllability of invariant control systems on $G$ in terms of the largest eigenvalues of second order differential operators. Keywords: Semisimple Lie groups, semigroups, moment Lyapunov exponent, flag manifolds. MSC: 22E46, 34D08, 22F30. [ Fulltextpdf (230 KB)] for subscribers only. 