Computational evidence for mild dissipation of the Hénon map at (1.4,0.3)

Crovisier and Pujals proved that Cr surface diffeomorphisms with |det Df|< 1/4 are mildly dissipative: their local stable manifolds separate the trappin gregion, enabling a reduction to one-dimensional dynamics. The classical Hénon map H(x,y) = (1−1.4x2 + y, 0.3x) has |det DH|= 0.3, just above this threshold. We present a computer-assisted proof that all 13,630,199 periodic orbits of periods 13–42 (found by kneading-constrained exhaustive search) satisfy the Wiman condition λ+ ≥0.198824 > log(4b) ≈0.1823. The global min- imum is achieved by a unique period-13 orbit γ13 with λ+ = 0.198824. Combining this with Katok’s closing lemma and Sarig’s symbolic dynamics, we prove unconditionally that every ergodic measure with entropy h(µ) ≥0.0165 satisfies the Wiman condition, and con- ditionally (assuming the bound holds for all periodic orbits) that every ergodic invariant measure satisfies it. An additional 1,526 orbits at periods 38–100 (mpmath, 50 digits) all pass with zero violations and asymptotic min λ+ →0.324. Independent verification by Julia IntervalArithmetic.jl confirms all results.