The Variational Principle for a Z^N_+ Action on a Hausdorff Locally Compact Space

The paper proves a variational principle for topological pressure on locally compact Hausdorff spaces for Z_+^N-actions using admissible covers, including the normalization for one-point uniformly continuous potentials and the no-invariant-measure case (Theorem 4.1) . It establishes the cover-based pressure and shows independence from the chosen universal subnet and the equivalence with separated/spanning definitions (Lemma 3.5, Proposition 3.11) , and proves both inequalities: P_µ(T,f) ≤ P(T,f) (Proposition 4.2) and, for each admissible cover A, existence of an invariant measure µ with S(T,f,A) ≤ P_µ(T,f) (Proposition 4.3) . The candidate solution reaches the same conclusion via a slightly different Misiurewicz-style construction (weights on near-optimal subcovers rather than the paper’s separated-set/compact-extension route). Minor slips in the candidate text (e.g., a “taking inf over µ” typo and a loose tightness remark) do not affect the core argument. Hence, both are correct, with substantively different implementations of the standard compact-case template adapted to admissible covers.