On the shadowing properties for $\mathscr{G}$-processes

This paper is devoted to the investigation of several notions associated with the shadowing behavior of ${\mathscr{G}}$-processes. We introduce and explore various types of shadowing, including standard shadowing, $h$-shadowing, limit shadowing, $s$-limit shadowing, and exponential limit shadowing for ${\mathscr{G}}$-processes. It is shown that if two ${\mathscr{G}}$-processes are uniformly conjugate, then the presence of any of these shadowing properties in one implies the same property holds for the other. Moreover, we establish that a finite direct product of ${\mathscr{G}}$-processes possesses the $h$-shadowing property if and only if each constituent ${\mathscr{G}}$-process in the product exhibits this property. The same equivalence also holds for the other types of shadowing considered.