Renormalisation of the two-dimensional border-collision normal form.

The paper proves that, on the preimage Πξ of the right half-plane, the two-step map f^2_ξ has exactly two affine pieces and is affinely conjugate to the same border-collision normal form with parameters g(ξ) = (τ_R^2−2δ_R, δ_R^2, τ_L τ_R−δ_L−δ_R, δ_L δ_R) (see (2.9)–(2.10) and Proposition 4.1 with the explicit h_ξ in (4.1), together with f^2_ξ = h_ξ^{-1} ∘ f_{g(ξ)} ∘ h_ξ on Πξ) . It then organizes parameter space via ζ_n(ξ)=φ(g^n(ξ)) and regions R_n:={ζ_n>0, ζ_{n+1}≤0} (2.12)–(2.13), proves Theorem 2.2 for ξ∈R_0 with Λ(ξ)=cl(W^u(X)) (2.15), and bootstraps by renormalisation to Theorem 2.3 (substitution (L,R)→(RR,LR); existence of 2^n disjoint sets S_i permuted by f_ξ; and affine conjugacy of f^{2^n}_ξ|_{S_i} to f_{g^n(ξ)}|_{Λ(g^n(ξ))}) (2.17)–(2.19) . The candidate reproduces this renormalisation scheme almost verbatim: they identify the same two-branch decomposition of f^2_ξ on the R-first-return domain, construct an explicit affine normalizer H_R whose effect matches h_ξ (yielding the same g), and carry out the same induction, including the symbolic substitution and union-of-unstable-manifold description. Minor differences are cosmetic: their H_R = S(−f_R)+t is an explicit affine realisation equivalent to the paper’s h_ξ, and they describe the domain as a R-side first return instead of Πξ. One slight overstatement is calling φ(ξ)>0 “exactly the nondegeneracy” of the two-branch return on Λ(ξ): in the paper φ>0 is the no-homoclinic-corner condition defining Φ_BYG, while the two-piece structure on Πξ holds for all ξ∈Φ; but for the R_0/R_n constructions the candidate’s usage aligns with (2.12)–(2.13) and the ensuing proofs. Overall, the model’s argument tracks the paper’s logic and results closely and correctly.