GARDEN OF EDEN AND WEAKLY PERIODIC POINTS FOR CERTAIN EXPANSIVE ACTIONS OF GROUPS

The paper proves that for expansive actions of countable amenable groups with weak specification and the strong topological Markov property (STMP), the Moore–Myhill (Garden of Eden) theorem holds; it gives a tiling–gluing–entropy proof of Moore (surjective ⇒ pre-injective) using STMP to glue local prescriptions and an entropy drop via a tailored closed set Z, and re-proves Myhill (pre-injective ⇒ surjective) under weak specification alone . The candidate solution follows the same strategy: finite-memory, (A,B)-tiling, STMP gluing to get τ[Z]=X, and an entropy contradiction, plus the standard weak-specification argument for Myhill. Minor issues: (i) the (A,B)-tiling was stated with A=PE, B=E (should satisfy A⊆B; reversing fixes it), and (ii) Step 2 erroneously concludes min{d(z,t^{-1}x), d(z,t^{-1}y)}≥δ/8, whereas the paper only needs and proves d(z,t^{-1}x)≥δ/8 . These do not alter the core logic. Overall, both are correct and essentially the same proof skeleton.