Nonexistence of observable chaos and its robustness in strongly monotone dynamical systems

For strongly monotone dynamical systems on a Banach space, we show that the largest Lyapunov exponent $\lambda_{\max}>0$ holds on a shy set in the measure-theoretic sense. This exhibits that strongly monotone dynamical systems admit no observable chaos, the notion of which was formulated by L.S. Young. We further show that such phenomenon of no observable chaos is robust under the $C^1$-perturbation of the systems.