Making rare events typical in d-dimensional chaotic maps

The paper proves three pillars: (i) the generalized Doob generator LD_s0 built from the tilted Frobenius–Perron operator preserves probability and has invariant density ρ_s0 = l_s0 r_s0 (Eq. (26) and discussion) ; (ii) there exists a coordinate transform γ_s0 (constructed via the Rosenblatt transform) pushing ρ to ρ_s0 so that the Doob effective map f^D = γ_s0 ∘ f ∘ γ_s0^{-1} is topologically conjugate to f ; and (iii) the transformed dynamics typicalizes the s-ensemble and shifts the LDP by I_D_s0(a) = I(a) + s0 a + θ(s0) (derived from the tilted distribution) . The candidate solution reproduces these results: Part A establishes (LD_s0)†[1]=1 and LD_s0[ρ_s0]=ρ_s0 directly from the eigen-relations, exactly as in the paper ; Part B constructs γ_s0 via Rosenblatt and proves the conjugacy f^D = γ_s0 ∘ f ∘ γ_s0^{-1} and ergodicity preservation, matching the paper’s statements and construction (Eqs. (36)–(42)) ; Part C derives the rate-function shift and, equivalently, the SCGF shift θ(s0+σ)−θ(s0), in line with the paper’s rate-shift claim . One extra observation in the candidate solution—namely that LD_s0 reduces to the original Frobenius–Perron operator L for deterministic maps—follows immediately from the left-eigenfunction identity and the δ-kernel, and is consistent with the paper’s framework (though not explicitly remarked there). The only caveat is a wording in the candidate’s remark suggesting multiple invariant absolutely continuous measures; for standard expanding maps the unique ACIM property holds, so ρ_s0 may be singular—something the paper also acknowledges in practice via fractal left eigenfunctions and discretization remarks .