Multiplicative ergodic theorem of semi-discrete dynamic system

In this paper, Multiplicative Ergodic Theorem (MET) on manifolds with semi-discrete time variable is proved. Considering that there is no cocycle property with any semi-discrete time variable t\in \mathbb {T}, we define the quasi-cocycle property on forward and backward time scales. We obtain the skew-product quasi-flow with semi-discrete time variable t\in \mathbb {T}. For dynamic equations with \Delta-derivative and \nabla-derivative on \mathbb {T}, we present a more generalized version about MET of semi-discrete system. The result is more suitable for treating models with nonuniform time difference and studying the stability of systems induced by both differential and difference operators.