Observable Lyapunov irregular sets for planar piecewise expanding maps

The paper proves two claims. First, for each integer r with 1 ≤ r < ∞, it constructs a one-parameter family of planar piecewise C^r expanding maps Fσ with a wandering rectangle Rσ such that, for Lebesgue-a.e. x ∈ Rσ and all v ≠ 0, the Lyapunov growth-rate ω-limit set ω(AFσ(x,v)) is a nontrivial interval that collapses to a point as σ → 0, with a uniform positive lower bound on its infimum; this is Theorem A and follows from an explicit non-conformal construction (Sections 2–3) and exact calculations yielding ω(AF(x,v)) = [ξ1, ξ0] for a.e. x in the wandering set . Second, in strong contrast, for piecewise real-analytic expanding planar maps, any Lyapunov irregular set has Lebesgue measure zero (Theorem B), proved via a spectral decomposition with a finite-hitting-time property to supports of finitely many ergodic acips, then Oseledets and a finite-prefix argument to transfer exponents to Lebesgue-a.e. initial point . The model states the same conclusions but its arguments are flawed: (A) it invokes a two-lane, near-conformal “coding”/independence mechanism that is neither used nor needed and not justified in the paper’s non-conformal setting; the paper instead derives the interval [ξ1, ξ0] via exact iteration formulas. (B) it asserts that basins of acips cover Lebesgue-a.e. points and then directly applies Oseledets; this misses the crucial finite-hitting-time step and the transfer-of-exponents argument that the paper provides. Hence the paper is correct; the model’s solution is not a correct proof.