FINITNESS OF MEASURED HOMOCLINIC CLASSES WITH LARGE LYAPUNOV EXPONENTS FOR C2 SURFACE DIFFEOMORPHISMS

The paper proves the finiteness result at the 19/20·R(f) Lyapunov-exponent threshold via a new, quantitative, CP-hyperbolic-set argument using Pliss’ lemma and the Crovisier–Pujals stable manifold theorem, followed by a compact covering/homoclinic-intersection argument. This matches its stated Theorem 1.1 and Theorem 4.1 and is internally consistent. By contrast, the model’s Step 3 incorrectly attributes to Buzzi–Crovisier–Sarig a finiteness statement for irreducible symbolic components under a Lyapunov-exponent threshold χ > 19/20·R(f); the 2022 Buzzi–Crovisier–Sarig result controls finiteness under an entropy threshold h > R(f)/r, not an exponent threshold. The model neither cites nor uses Ruelle’s inequality to bridge exponents to entropy. Hence the model’s proof, as written, relies on a non-existent input and is not correct.