Discrete dynamical systems in \(L\)-topological spaces
The study is focused on an analysis of some properties of discrete dynamic systems defined in \(L\)-topological spaces. The description of such systems is based upon the definitions of \(L\)-image and \(L\)-preimage operators, \[ \begin{aligned} f^\rightarrow(A)(y) &=\vee\{A(t): t\in X, f(t)= y\}\,A\in L^X,\;y\in Y,\\ f^\leftarrow(B)(x) &= B(f(x))\,B\in L^Y,\;x\in X,\end{aligned} \] where \(L\) denotes a completely distributive lattice. The family \(\{f^{\rightarrow n},n\in\mathbb{Z}\}\) defines a discrete dynamical system in \((L^X,\delta)\) with \(\delta\) being the \(L\)-topology on \(X\). (\(f^{\rightarrow-n}= f^{\leftarrow n}\), \(n= 0,1,2,\dots\) assuming that \(f\) is a homeomorphism). Studied are the key issues of such dynamical systems, namely their topological mixing, transitivity, and minimality. In the sequel, these aspects are compared with their counterparts existing in nonfuzzy spaces.
scientific article; zbMATH DE number 5001333
scientific article