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Keywords:
weakly fuzzy compact; weakly fuzzy compact topological dynamical system; weakly fuzzy topological entropy
weakly fuzzy compact; weakly fuzzy compact topological dynamical system; weakly fuzzy topological entropy
Summary:
In 2005, İ. Tok fuzzified the notion of the topological entropy R. A. Adler et al. (1965) using the notion of fuzzy compactness of C. L. Chang (1968). In the present paper, we have proposed a new definition of the fuzzy topological entropy of fuzzy continuous mapping, namely weakly fuzzy topological entropy based on the notion of weak fuzzy compactness due to R. Lowen (1976) along with its several properties. We have shown that the topological entropy R. A. Adler et al. (1965) of continuous mapping $\psi \colon (X,\tau )\rightarrow (X,\tau )$, where $(X,\tau )$ is compact, is equal to the weakly fuzzy topological entropy of $\psi \colon (X,\omega (\tau ))\rightarrow (X,\omega (\tau ))$. We have also established an example that shows that the fuzzy topological entropy of İ. Tok (2005) cannot give such a bridge result to the topological entropy of Adler et al. (1965). Moreover, our definition of the weakly fuzzy topological entropy can be applied to find the topological entropy (namely weakly fuzzy topological entropy $h_w(\psi )$) of the mapping $\psi \colon X\rightarrow X$ (where $X$ is either compact or weakly fuzzy compact), whereas the topological entropy $h_a(\psi )$ of Adler does not exist for the mapping $\psi \colon X\rightarrow X$ (where $X$ is non-compact weakly fuzzy compact). Finally, a product theorem for the weakly fuzzy topological entropy has been established.
References:
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