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Journal of Lie Theory 30 (2020), No. 2, 587--616
Copyright Heldermann Verlag 2020

Semigroups and Moment Lyapunov Exponents

Luiz A. B. San Martin
Universidade Estadual de Campinas, Campinas - SP, Brazil


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.

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