Nonwandering sets and the entropy of local homeomorphisms

The paper rigorously proves (i) hd(σ) = max{hd(σ|Ω), hd(σ|W̄)} via restriction to closed invariant pieces and (ii) on compact spaces with clopen domain, hd(σ) = hd(σ|Ω) using a nontrivial generating-set/dynamical-ball argument. The candidate solution’s Part (1) is essentially correct (it recovers the max formula by a subadditivity argument on separated sets). However, Part (2) asserts the stronger claim that the wandering part has zero entropy and bases this on an unjustified linear bound for the size of an n-fold joined cover on a wandering open set. That linear-growth claim is incorrect in general: the join cover can grow exponentially even when forward images of a wandering neighborhood are disjoint, and the direction of refinement used to compare covers is reversed. The paper does not claim hd(σ|W) = 0; it proves hd(σ) ≤ hd(σ|Ω) and hence equality, while only deducing the inequality hd(σ|W) ≤ hd(σ|Ω).