Li-Yorke chaos on fuzzy dynamical systems

Given a dynamical system we investigate how several variants of Li-Yorke chaos behave with respect to the extended systems and , where is the hyperextension of acting on the space of non-empty compact subsets of , and where denotes the Zadeh extension of acting on the space of normal fuzzy subsets of . We first prove that the main variants of Li-Yorke chaos transfer from to and from to , but that the converse implications do not hold in general. However, combining the notions of proximality and sensitivity we introduce Cantor-dense Li-Yorke chaos, and we prove that this strengthened variant of chaos does transfer from to under natural assumptions.