Stable Lyapunov Spectrum Rigidity of Nilmanifold Endomorphisms

The paper proves, in the codimension‑one stable setting with totally non‑invertible and horizontally irreducible linear part Ψ, the equivalence among: (1) existence of an invariant unstable bundle, (2) topological conjugacy to Ψ, (3) constancy of periodic stable exponents, (4) equality of those exponents with Ψ, and (5) Cr* regularity of the unstable bundle, and further shows any conjugacy is Cr+1 along stable leaves when f is Cr+1 (Cor. 1.8) . The candidate’s chain of implications matches the paper’s: (2)⇒(3) via Theorem 4.1 using exponential density of preimages for totally non‑invertible Ψ (Thm. 3.4) , (3)⇒(4) via Theorem 2.27 , (3)⇒(2) under horizontal irreducibility via Theorem 4.9 , and (1)⇔(2) in the 1‑dimensional stable case by Theorem 2.28 plus Remark 2.9 (u‑ideal holds) . Leafwise smoothness of the conjugacy on stable leaves follows from Theorem 4.2 and Corollary 1.4 (and is upgraded to Cr+1 in Cor. 1.8 under f ∈ Cr+1) . The candidate’s references to a Livšic argument and to projecting the conjugacy agree with the paper’s use of Livšic (Prop. 4.4) to build the affine metric ds and obtain leafwise smoothness (Theorem 4.2) and projection (Prop. 2.29) . Minor differences are expository (e.g., the candidate mentions “normal forms,” while the paper closes (4)⇒(5) via holonomy regularity using ds and PSW97) .