Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems

Let X be a compact metric space and a continuous map f:X→X which defines a discrete dynamical system (X,f). The map f induces two natural maps, namely f¯:K(X)→K(X) on the hyperspace K(X) of non-empty compact subspaces of X and the Zadeh’s extension f^:F(X)→F(X) on the space F(X) of normal fuzzy set. In this work, we analyze the interaction of some orbit tracing dynamical properties, namely the specification and shadowing properties of the discrete dynamical system (X,f) and its induced discrete dynamical systems (K(X),f¯) and (F(X),f^). Adding an algebraic structure yields stronger conclusions, and we obtain a full characterization of the specification property in the hyperspace, in the fuzzy space, and in the phase space X if we assume that the later is a convex compact subset of a (metrizable and complete) locally convex space and f is a linear operator.