Multi-sensitivity, syndetical sensitivity and the asymptotic averageshadowing property for continuous semi-flows

In this paper, for a continuous semi-flow θ on a compact metric space E with the asymptotic average-shadowing property (AASP), we show that if the almost periodic points of θ are dense in E then θ is multi-sensitive and syndetically sensitive. Also, we show that if θ is a Lyapunov stable semi-flow with the AASP, then the space E is trivial. Consequently, a Lyapunov stable semi-flow with the AASP is minimal. Furthermore, we prove that for a syndetically transitive continuous semi-flow on a compact metric space, sensitivity is equivalent to syndetical sensitivity. As an application, we show that for a continuous semi-flow θ on a compact metric space E with the AASP, if the almost periodic points of ϕ are dense in E then θ is syndetically sensitive. Moreover, we prove that for any continuous semi-flow θ on a compact metric space, it has the AASP if and only if so does its inverse limit (E~, θ~), and if only if so does its lifting continuous semi-flow (Eˆ, θˆ). Also, an example which contains two numerical experiments is given. Our results extend some corresponding and existing ones.