Narrow Results

Let $(X,f)$ be a dynamical system, $({𝒦}(X),\overline{f})$ be the system on hyperspace and $(𝔽(X),{\tilde{f}}_g)$ be its $g$-fuzzification. We study the relations among the various forms of sensitivity of the systems $(X,f)$, $({𝒦}(X),\overline{f})$ and $({𝔽}^1(X),{\tilde{f}}_g)$. Firstly, we give a simpler proof of the theorem in [Wu & Chen, 2017] which states that $({𝒦}(X),\overline{f})$ is sensitive if and only if $({𝔽}^1(X),{\tilde{f}}_g)$ is sensitive. Then, we show that (1) all forms of sensitivity of $({𝒦}(X),\overline{f})$ are the same for $(𝔽(X),{\tilde{f}}_g)$, (2) all forms of sensitivity of $({𝒦}(X),\overline{f})$ partly imply the same for $(X,f)$, and the converse holds in some cases.

In this paper by using the relative fuzzy topology a new kind of dynamics is considered. A method for constructing relative semi-dynamical systems is presented. Relative fixed points and relative sinks are also studied. New results on invariant sets without any similarity in usual mathematics are deduced. Furthermore an extension of the notion of conjugate relation is presented.

Within this contribution we establish a theoretical background necessary for studying inverse limits of fuzzy dynamical systems induced by crisp (non-fuzzy) ones. First, we elaborate topological properties of the space of fuzzy sets, such as connectedness and compactness. Second, we focus our attention on dynamical conditions sufficient for the existence of indecomposable continua in the inverse limit space.