Shadowing and mixing on systems of countable group actions

The paper proves that for a countable discrete group action on a compact totally disconnected Hausdorff space, Φ has S-shadowing iff the system is conjugate to an ML inverse limit of shifts of finite type over A = S ∪ {e_G} (Theorem 1.1). The backward direction uses that SFTs over A have S-shadowing and that ML inverse limits preserve S-shadowing (Theorem 3.4), while the forward direction encodes dynamics via pseudo-orbit spaces POS(U), shows POS(U) are SFTs over A, uses S-shadowing to get image stabilization ι(POS(W)) = O(U) (Lemma 4.3), and concludes conjugacy to an ML inverse limit of SFTs (Theorem 4.4, Corollary 4.5) . The candidate solution reconstructs the same scheme with clopen partitions: builds SFTs X_λ over A from local A-block constraints (equivalent to the paper’s POS(U)), defines refinement bonding maps, derives the ML condition via S-shadowing (stabilization of images to the orbit code O(U)), and obtains a conjugacy by a standard coding map. Thus, the model’s proof aligns closely with the paper’s proofs, differing mainly in presentation and notation.