The Metric Entropy of Endomorphisms

We prove that, for a C ² non-invertible but non-degenerate map on a compact Riemannian manifold without boundary, an invariant measure satisfies an equality relating entropy, folding entropy and negative Lyapunov exponents. This generalizes Ledrappier-Young’s entropy formula 5] (for negative Lyapunov exponents of diffeomorphisms) to the case of endomorphisms. This work is supported by National Basic Research Program of China (973 Program) (2007 CB 814800).