Renormalization in Lorenz Maps - Completely Invariant Sets and Periodic Orbits

The candidate solution proves exactly the four assertions of Theorem 3.5 for expanding Lorenz maps with a primary n(k)-cycle, using a clean “gap-dynamics” lemma with modular arithmetic on the n-cycle to control first-hitting times of c and to build the renormalization g = f^n on [u,v]. The paper’s Theorem 3.5 states the same four items and provides a (somewhat different) proof: (1) construction of g on [u,v]=[f^{n-1}(0),f^{n-1}(1)] and continuity/monotonicity of f^n on [u,c),(c,v], (2) divisibility of l,r by n for any renormalization with l or r ≥ n, (3) non-complete invariance of F̂_g in the endpoint-equality cases, and (4) complete invariance and boundary identification for J_g under strict inequalities, with R_{J_g}f = g. See Theorem 3.5 statement and proof in the paper, where items (1) and (4) are developed via interval-mapping arguments (e.g., (3.4)–(3.6) and the analysis around u=f^{n-1}(0), v=f^{n-1}(1)), and the first-hitting time n is identified via Lemma 4.11/Equation (4.3) (N((π(P̂_L),c))=N((c,π(P̂_R)))=n), which underlies both (2) and boundary claims in (4b) (the proof concludes e−=z_{n−k−1}, e+=z_{n−k}). The candidate’s approach is sound and aligns with the paper’s conclusions, differing mainly in technique (a direct iterative interval-tracking with a gcd(k,n)=1 congruence argument vs. the paper’s two-case structure and use of the doubled system and hitting-time lemmas). Consequently, both are correct, with different proofs. Key references: Theorem 3.5 statement and overall structure, including (1)–(4) ; continuity/monotonicity and placement of u,v in the central gap for (1) ; first-hitting time n (Equation (4.3)) ; identification of e−,e+ and completion of (4b,c) .