Genericity of Lyapunov Spectrum of Random Dynamical Systems

Our first aim in this chapter is to prove genericity of analyticity of Lyapunov spectrum for discrete-time bounded linear random dynamical systems and bounded linear random differential equations. The main ingredient in the proof is the openness and denseness of integral separations for these bounded linear random dynamical systems. This property is no longer true for unbounded linear random dynamical systems. So, our second aim in this chapter is to construct an open set of unbounded linear random dynamical systems with simple Lyapunov spectrum but having no exponential separated splitting. This example leads to several open questions on the genericity of Lyapunov spectrum of unbounded linear random dynamical systems. The materials of this chapter are taken from Cong and Doan (Stoch Dyn. 7(3):335–355, 2007), (Discrete Contin Dyn Syst Seri S 9:995–1007, 2016), Doan (Discrete Contin Dyn Syst B 22(8):3113–3126, 2017).