The first results about the concepts of pseudotrajectory and shadowing property can be found in the works of Bowen, Conley, and Sinai. Since their introduction, these concepts have been extensively investigated by many authors in various contexts. Applications of these notions can be found in many areas of the theory of dynamical systems.
C. Good and J. Meddaugh [Invent. Math. 220, No. 3, 715–736 (2020; Zbl 1445.37015)] introduced shadowing for continuous maps on arbitrary compact Hausdorff spaces and found a fundamental structural connection between the shadowing property and inverse limits of inverse systems satisfying the Mittag-Leffler condition and consisting of shifts of finite type. U. B. Darji et al. [Adv. Math. 385, Article ID 107760, 34 p. (2021; Zbl 1471.37023)] established fundamental connections between shadowing, finite-order shifts, and ultrametric complete spaces. Among other things, they found connections between the shadowing property in general metric spaces and inverse limits of inverse systems satisfying the Mittag-Leffler condition and consisting of shifts of finite order for infinite alphabets.
Motivated by these results, the author proves the following theorem (Theorem 4 in the paper):
Theorem. Let \(((X_{\alpha},T_{\alpha}),f_{\alpha \beta})\) be an inverse system of dynamical systems on uniform spaces with index set \(I\) and let \((X, T)\) be its inverse limit. Suppose that each \((X_{\alpha},T_{\alpha})\) has the (finite) shadowing property. If for every \(\alpha \in I\), there exists \(\beta \geq \alpha\) such that \(f_{\alpha \beta}(X_{\beta}) \subset f_{\alpha}(X_\alpha)\), then \((X, T)\) has the (finite) shadowing property. In particular, if the Mittag-Leffler condition holds and if

(i)

\(I\) contains a countable cofinal set, or

(ii)

Each \(f_{\alpha \beta}\) is injective, or

(iii)

Each \(X_{\alpha}\) is a compact Hausdorff space and each \(T_{\alpha}\) is a continuous map,

then \((X, T)\) has the (finite) shadowing property.
The notion of pseudotrajectories is also related to other concepts. In this way, the author establishes results similar to the above theorem, where the shadowing property is replaced by chain recurrence, chain transitivity, total chain transitivity, and chain mixing (see Theorems 11 and 13).
In the last part of the paper, the following variations of shadowing are examined: the eventual shadowing property, the orbital shadowing property, the strong orbital shadowing property, the first weak shadowing property, and the second weak shadowing property. All these variations of shadowing are covered by a unifying theorem (see Theorem 15).