Abstract
BACKGROUND: Shadowing-type properties play a fundamental role in the qualitative theory of dynamical systems, as they describe the relationship between approximate trajectories and exact orbits. In recent years, increasing attention has been given to extending these concepts to set-valued mappings, which naturally arise in various areas of mathematics and applied sciences. However, several shadowing-related notions for such mappings remain insufficiently explored. METHODS: In this work, we introduce precise definitions of the inverse shadowing property and the ergodic shadowing property for set-valued mappings. We analyse these properties within a general topological framework and examine their behaviour under the shift mapping on the inverse limit space. The relationships between inverse shadowing and ergodic shadowing are investigated using tools from topological dynamics. RESULTS: We establish connections between the inverse shadowing property and the ergodic shadowing property for set-valued mappings. In particular, we show how these properties interact when considered together with the shift mapping on the inverse limit space, and we identify conditions under which one property implies the other. CONCLUSIONS: The results provide a clearer understanding of shadowing phenomena for set-valued mappings and highlight the role of inverse limit spaces in studying their dynamical behavior. This work contributes to the development of shadowing theory beyond single-valued dynamics and offers a foundation for further investigations in this direction.