Expansive properties of induced dynamical systems

Abstract

For a given metric space X, we consider the set of all normal fuzzy sets on X, denoted by F(X). In this work, we study expanding, positively expansive and weakly positively expansive dynamical systems (X,f) and how they are reflected in the dynamical system (F(X),fˆ), where fˆ is the Zadeh's extension of f and F(X) has one of the following metrics: the levelwise metric, the endograph metric, the sendograph metric and the Skorokhod metric. We mainly show that if we consider the following conditions:(i)(X,f) is positively expansive (resp. expanding);(ii)(K(X),f‾) is positively expansive (resp. expanding);(iii)(F∞(X),fˆ) is positively expansive (resp. expanding);(iv)(F0(X),fˆ) is positively expansive (resp. expanding). Then (iv)⇒ (iii) ⇔ (ii) ⇒ (i). For expanding dynamical systems, we present a compact metric space and a locally compact metric space to show that (i) ⇏ (ii) and (iii) ⇏ (iv), respectively. For positively expansive dynamical systems, there is a compact metric space satisfying that (i) ⇏ (ii), but we don't know if (iii) ⇒ (iv).