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

Abstract

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 ( E, θ). Also, an example which contains two numerical experiments is given. Our results extend some corresponding and existing ones.

Weak mixing in general semiflows implies multi-sensitivity, but not thick sensitivity

“…We thus generalized Moothathu's first comment to the case of general semiflows and showed that his second comment does not hold in general semiflows. Let us also mention that multi-sensitivity for continuous semiflows was discussed in [5].…”