Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 30 (2020), No. 2, 587--616Copyright Heldermann Verlag 2020 Semigroups and Moment Lyapunov Exponents Luiz A. B. San Martin Universidade Estadual de Campinas, Campinas - SP, Brazil smartin@ime.unicamp.br [Abstract-pdf] Let $G$ be a noncompact semi-simple 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: Semi-simple Lie groups, semigroups, moment Lyapunov exponent, flag manifolds. MSC: 22E46, 34D08, 22F30. [ Fulltext-pdf  (230  KB)] for subscribers only.