Shadowing, average shadowing and transitive properties of multiple mappings

In 2016, Hou and Wang introduced the concept of multiple mappings based on iterated function system, which is an important branch of fractal theory. In this paper, we introduce the definitions of shadowing, average shadowing, transitive, weakly mixing, mixing, chain transitive and chain mixing properties of multiple mappings from a set-valued perspective. We show that both two continuous self-maps have shadowing (respectively, average shadowing, transitive, weakly mixing, mixing) property may not imply the multiple mappings they form has the corresponding property. While a sufficient condition for multiple mappings to be transitive (respective, weakly mixing, mixing, chain transitive, chain mixing) is given. Both shadowing and average shadowing properties are invariant under the iterative action of multiple mappings. Also, we study that for multiple mappings shadowing property plus chain mixing implies mixing and average shadowing properties implies chain transitivity.