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On Ergodic and Inverse Shadowing Properties of Set-Valued Mapping
Farah Wattan Kamil1, Iftichar M T Al-Shara'a1
1Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Babil, Iraq.
F1000Research
|April 23, 2026
Summary
This study defines inverse and ergodic shadowing properties for set-valued mappings, crucial for understanding dynamical systems. The research clarifies their relationships within topological dynamics, advancing shadowing theory beyond single-valued systems.
Area of Science:
- Dynamical Systems Theory
- Set-Valued Mappings
- Topological Dynamics
Background:
- Shadowing properties are key in dynamical systems, linking approximate and exact trajectories.
- Extending shadowing concepts to set-valued mappings is an emerging research area.
- Existing shadowing notions for set-valued mappings are not fully explored.
Purpose of the Study:
- Introduce precise definitions for inverse shadowing and ergodic shadowing properties for set-valued mappings.
- Analyze these properties within a general topological framework.
- Investigate the behavior of these properties under shift mapping on inverse limit spaces.
Main Methods:
- Utilized tools from topological dynamics to analyze set-valued mappings.
- Examined properties within a general topological framework.
- Investigated interactions with shift mapping on inverse limit spaces.
Main Results:
- Established connections between inverse shadowing and ergodic shadowing properties for set-valued mappings.
- Demonstrated how these properties interact with shift mapping on inverse limit spaces.
- Identified conditions for one property to imply the other.
Conclusions:
- Enhanced understanding of shadowing phenomena in set-valued mappings.
- Highlighted the significance of inverse limit spaces in analyzing dynamical behavior.
- Contributed to the advancement of shadowing theory for non-single-valued dynamics.
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