Invariant Measures Satisfying an Equality Relating Entropy, Folding Entropy and Negative Lyapunov Exponents

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In this paper we prove that, for a C2 (non-invertible but non-degenerate) map on a compact manifold, an invariant measure satisfies an equality relating entropy, folding entropy and negative Lyapunov exponents if and, under a condition on the Jacobian of the map, only if the measure has absolutely continuous conditional measures on the stable manifolds.