Invariant graphs and dynamics of a family of continuous piecewise linear planar maps

The paper’s Theorem D states exactly that, after reducing to one essential parameter c = -b/a for a<0, the entropy of F|Γ is positive iff c ∈ (α, -1/36) ∪ (β, 1) ∪ (1, 8), with α ∈ (-112/137, -13/16), β ∈ (603/874, 563/816), is non-decreasing in c on the first two intervals, and is discontinuous at c = -1/36; this is proved via (i) construction of the invariant graph Γ (Theorem B) and (ii) two trapezoidal unimodal semiconjugacies, plus explicit “rome-method” Markov computations in the remaining window, exactly as summarized by the candidate solution . The candidate’s argument follows the same structure and ingredients (scaling conjugacy, Γ-graph reduction, trapezoidal factors with monotone entropy, and rome-based SFT lower bounds) and reaches the same conclusions. Minor slips include the set of edge slopes on Γ (paper: 0, ±1, 8; candidate wrote 0, ±1, 2) and a mis-citation around degree-one circle maps. Otherwise, the logic, reductions, monotonicity mechanism, and window identifications match the paper’s proof strategy, and the parameter brackets α, β coincide with those in the paper .