Invariant measures for interval maps without Lyapunov exponents

The paper’s Theorem 1 states exactly the four items the model proves: (1) the metric Lyapunov exponent for μ̃_P is not defined; (2) pointwise Lyapunov exponents on Of(c̃) do not exist when the orbit avoids S(f), and are undefined when it hits S(f); (3) log dist(·,S(f)) is not integrable with respect to μ̃_P; and (4) the orbit of the Lorenz-like singularity has exponential recurrence . The paper establishes the Lorenz-like singularity of f via non-flatness of h (Proposition 1.1) and precise comparisons |f′(x)| ≍ |x−c̃|^{-ℓ±}, with ℓ± = α±/(α±+1) , proves non-integrability of both the positive and negative parts of log|f′| and of |log dist(·,S(f))| (Proposition 1.2) using a telescoping identity and a Fibonacci/Sturmian measure estimate , shows pointwise exponent oscillation on Of(c̃) (Proposition 1.3) , and quantifies exponential recurrence (Proposition 1.4) . The model’s solution mirrors these ingredients: (i) conjugacy to the Fibonacci tent map and a telescoping identity; (ii) extraction of Lorenz-like orders from the non-flatness of h; (iii) use of Fibonacci returns and Sturmian combinatorics to estimate masses and distances; and (iv) deduction of non-integrability and non-existence of Lyapunov exponents. One minor slip is the claim in Step 3 that the limsup of (1/n)log|(f^n)'(x)| is strictly below log λ; the paper correctly shows limsup ≥ log λ and liminf < log λ (so the limit does not exist) . Aside from this, the proofs are essentially the same in structure and content.