Every compact Hausdorff dynamical system is a factor of an inverse limit of shifts of finite type; shadowing ensures surjective bonding maps and Mittag-Leffler inverse sequences in the metric case.
Doucha, Strong topological Rokhlin property, shadowing, and symbolic dynamics of count- able groups, J
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Shadowing in Dynamical Systems: Zero-dimensional Extensions and Inverse Limits
Every compact Hausdorff dynamical system is a factor of an inverse limit of shifts of finite type; shadowing ensures surjective bonding maps and Mittag-Leffler inverse sequences in the metric case.