Transitivity and Topological Entropy on Fuzzy Dynamical Systems Through Fuzzy Observation

Any physical or geometrical variation on a natural dynamical system should be identified by an observer. Also a method is required to compare different observers and evaluate their perspectives. Moreover complexity and/or uncertainty of the system should be measured through viewpoint of observers. In the approach presented in this paper an observer is identified mathematically by a function μ : X → [0, 1], where X denotes the base space of the system; the μ-Fuzzy Topology is defined as a description of the topological notion on X by the eyes of the observer μ. This idea will be applied to the other physical and geometrical notions such as minimality, transitivity and topological entropy for a fuzzy dynamical system on X. It will suggest a rational description of uncertainty in natural systems. AMS Subject Classification: 37B99, 93A30 Key–Words: Relative Semi-dynamical System, Transitivity, Relative Topological Entropy