On Ergodic and Inverse Shadowing Properties of Set-Valued Mapping

This paper examines the correlation between the inverse shadowing property and the ergodic shadowing property for set-valued mappings. First, we present appropriate definitions of inverse shadowing and ergodic shadowing properties within the context of set-valued dynamical systems. We then examine these properties through the shift mapping σ_H on the inverse limit space related to the mapping. We derive several results elucidating the behavior of these shadowing properties under the induced dynamical system on the inverse limit space. We specifically delineate the conditions necessary for the preservation of the ergodic shadowing property in the shift mapping and examine its relationship with the inverse shadowing property in set-valued mappings. These findings broaden established correlations between shadowing properties and inverse limit spaces from single-valued dynamical systems to the field of set-valued dynamical systems.