Shadowing Maps

The paper’s Theorem 7.5 proves the equivalence between topological hyperbolicity (expansivity + shadowing) and the existence of a continuous shadowing map that is both shift-invariant and dynamically-invariant. The direction (1→2) constructs the canonical selector using uniqueness of shadowing under expansivity, yielding shift/dynamic invariance and continuity; this matches the model’s construction in spirit and detail. For (2→1), the paper deduces expansivity via an induced bracket with uniform contraction (Proposition 6.15 ⇒ Proposition 7.3 ⇒ Corollary 7.4), and shadowing via Proposition 3.2, whereas the model proves expansivity by a clean splice argument using shift/dynamic invariance and continuity of the selector. Both routes are correct; the proofs differ in technique but reach the same conclusion. See Theorem 7.5, its proof outline, and the supporting results on invariance and shadowing maps in the paper’s Sections 3 and 7 (Theorem 7.5; invariance via uniqueness noted in §1; shadowing map equivalence in Proposition 3.2; bracket contraction in Proposition 6.15; expansivity from bracket in Proposition 7.3 and Corollary 7.4).