Finite Chaoticity and Pairwise Sensitivity of a Strong-Mixing Measure-Preserving Semi-Flow

Chaos is a common phenomenon in nature and social sciences. As is well known, chaos has multiple definitions, and there are both differences and connections between them. The unique properties of chaotic systems can be leveraged to address challenges in communication, security, data processing, system analysis, and control across different domains. For semi-flows, this paper introduces two important concepts corresponding to discrete dynamical systems, finitely chaotic and pairwise sensitivity. Since Tent map and its induced suspended semi-flows both have these two properties, then these two concepts on the semi-flows have extensive and important applications and meanings in information security, finance, artificial intelligence and other fields. This paper extends the vast majority of corresponding results in discrete dynamical systems to semi-flows.