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

In this paper we prove that, for a C ² (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. This work is supported by National Basic Research Program of China (973 Program) (2007CB814800).