Dynamical Zeta Functions in the Nonorientable Case

The paper proves (i) meromorphic continuation of the Ruelle zeta for Axiom A flows without orientability assumptions by introducing a twisted trace/determinant identity on i_V=0 forms tensored with the orientation line o(E^s) and reducing to basic sets (Theorem 1, Proposition 1.4; see the stated factorization ζ_R = ∏ ζ_{K,k}^{(−1)^{k+dim E^s}}) and (ii) for 3D contact Anosov Reeb flows with nonorientable E^s, ord_{λ=0} ζ_R = b_1(o(E^s)) via Res^0(0)=Res^2(0)=0 and an explicit identification Res^1(0) ≅ H^1(M; o(E^s)) with semisimplicity at 0 proved directly (equations (18)–(19), Propositions 3.4, 3.6, 3.8–3.9) . The candidate solution reaches the same conclusions: (A) by spectral decomposition into basic sets and Pollicott’s meromorphy for each basic set, and (B) by the microlocal determinant identity, contact cancellations m_2(0)=m_0(0), H^0(M;o(E^s))=0, and the Dang–Rivière identification Res^1(0) ≅ H^1(M;o(E^s)). The proofs differ in method but agree on statements and key intermediate identities; no substantive logical conflicts were found.