EVEN THE VAGUE SPECIFICATION PROPERTY IMPLIES DENSITY OF ERGODIC MEASURES

The paper proves that for a surjective topological dynamical system with the vague specification property (equivalently, AASP), ergodic measures are weak* dense among invariant measures, by passing to the natural extension, approximating it via delta-chain subsystems that have the periodic specification property, establishing convergence in Besicovitch/\bar d senses, and using a stability result that preserves density in the limit, then transferring back via the natural extension’s affine homeomorphism of measure simplices. This is stated and executed in Proposition 7.1, Theorem 6.4, Proposition 2.16, Lemma 2.1, and culminates in Theorem 7.2 of the uploaded paper . The candidate solution mirrors this route almost verbatim: (i) reduce to the natural extension; (ii) use delta-chain models with periodic specification; (iii) use the AASP/VSP framework to obtain Besicovitch approximation; (iv) invoke convergence of measure simplices in a generalized \bar d and preservation of density; and (v) push back to (X,T). The only minor inaccuracy is the claim that each delta-chain model is itself chain mixing; what is needed (and what the paper proves) is that chain mixing of (X,T) is equivalent to periodic specification (and mixing) for every delta-chain approximant (Proposition 2.16) . This does not affect the argument. Overall, both are correct and essentially the same proof.