Shadowing property for set-valued map and its inverse limit
Zhengyu Yin
TL;DR
This work analyzes how the shadowing property $-$ also called pseudo-orbit tracing $-$ for set-valued maps $F:X\to 2^X$ relates to their inverse-limit constructions. It proves that if $F$ is expansive and open, then the corresponding inverse-limit system has the shadowing property, and that a continuous set-valued map has shadowing iff some induced inverse-limit system has it; moreover, the shadowing property of $F$ is equivalent to the shadowing of its induced inverse set-valued system $F_{inv}$. The results extend classical single-valued dynamics to the set-valued setting and provide a practical criterion for establishing shadowing via inverse limits.
Abstract
In this article, we investigate the relationship between the shadowing property of set-valued maps and their associated inverse limit systems. We show that if a set-valued map is expansive and open in the context of set-valued dynamics, then certain induced inverse limit systems have the shadowing property. Additionally, we prove that a continuous set-valued map has the shadowing property if and only if some of its induced inverse limit system also has shadowing property. Finally, we establish that the shadowing property of a set-valued map is equivalent to the shadowing property of its induced inverse set-valued system.