Generalized eigenvalues of the Perron-Frobenius operators of symbolic dynamical systems

The paper rigorously proves analytic continuation of the resolvent for VL(ε)=Q0+εQ1 and identifies generalized eigenvalues 2^{-k} with algebraic multiplicity k+1 and geometric multiplicity 1 for ε≠0, via the Gelfand triplet framework and a perturbative Neumann-series analysis. The model independently reaches the same conclusions using a finite-dimensional polynomial-core filtration and a triangular/Jordan-block analysis. While the model omits some functional-analytic details (e.g., passage to the inductive-limit topologies and continuity on X−→X′+), its argument correctly captures the mechanism and multiplicities. Hence both are correct, by different methods.