Year: 2021
Type: preprint
Abstract: For the family of deformed D_{2N} cluster maps we construct a single generating function I_N(λ) = Σ_k J_k λ^k of first integrals, valid for every N, as a transfer-matrix product over the reduced coordinates. We prove that (A) I_N(λ) is invariant under the map for every N, by exhibiting an explicit discrete zero-curvature (gauge) representation; (B) the integrals are in involution, by a finite invariant-subspace certificate for the transfer-matrix product that generates the Poisson bracket; (C) they are functionally independent, by computing the exact leading term of a degeneration; and (D) the map preserves the reduced log-canonical bracket, by a direct computation in a set of coordinates in which the bracket becomes a chain. Consequently the reduced map on the 2(N − 1)-dimensional symplectic leaf has N − 1 independent commuting integrals and is Liouville integrable, for every N ≥ 2. This proves the Liouville integrability of the reduced deformed D_{2N} maps, conjectured for every N on the basis of the algebraic entropy test [K25]; the case N = 2 is the deformed D₄ map of [HKM25]. Prepared with the help of an AI assistant (Claude Fable 5.1, Anthropic), which assisted in structuring the arguments, drafting the text and carrying out the exact symbolic and computer verifications; the mathematical content and the responsibility for it rest with the author. Source (Markdown) and PDF are both included.
Source: arXiv (Cornell University)
Institutions: University of Kent, University of Lincoln
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Award: G4536800330
OpenAlex ID: W4287068132