LIMIT SETS, INTERNAL CHAIN TRANSITIVITY AND ORBITAL SHADOWING OF TREE-SHIFTS DEFINED ON MARKOV-CAYLEY TREES

The paper’s Theorem 1.12 states that a tree-shift on a Markov-Cayley tree is TSFT if and only if it has the (projected) shadowing property, with precise definitions given in Definition 1.11 and the theorem statement itself . The forward implication constructs t|g = O(g)|ε and shows d(σg(t), O(g)) < ε (see the argument around equation (3.1)) . The reverse implication is proved contrapositively by building a δ-PPO that cannot be 1-shadowed when T is not TSFT (see (3.2)–(3.8)) . The candidate solution reproduces the forward direction essentially in the same spirit (defining t from root labels of O and using δ < 1/2 to propagate equality) and provides a different, constructive reverse direction via an R-language/gluing argument from a fixed ε0 = 1/2; this also correctly yields finite type. Hence both are correct, with substantially similar forward directions and a different, valid reverse proof.