A TRAJECTORY-DRIVEN ALGORITHM FOR DIFFERENTIATING SRB MEASURES ON UNSTABLE MANIFOLDS

The paper derives the key identity ρ̃(x) |det R(x)| = 1 from a measure-based parameterization of unstable manifolds (generalizing ρ̃‖x′‖ = 1 in 1D) and then differentiates it to obtain the SRB density-gradient formula g(i) in terms of the chart derivatives; after orthonormalization this reduces to g(i) = −Q(:j)·a(i,j) with the correct update recursions for first- and second-order derivatives via the chain rule and QR (Eqs. 4.3–4.5, 4.7–4.12) . The model’s Step A instead asserts that translating the parameter set V → V + t e_i leaves µ(x_k(V)) unchanged and concludes ∂_{ξ(i)}(ρ̃ J) = 0, which is false in general; the constancy ρ̃ J ≡ 1 holds only under the paper’s measure-based parameterization, not from translation invariance. Although the model’s later QR-based recursions and the 1D reduction match the paper’s formulas, the foundational derivation of g(i) lacks the crucial assumption and uses an incorrect argument, whereas the paper’s argument is internally consistent and aligns with its stated parameterization and updates (Eqs. 3.3, 4.3–4.5, 4.9–4.11, and the 1D formula Eq. 3.10) .