Dimension Theory for Invariant Measures of Endomorphisms

We establish the exact dimensional property of an ergodic hyperbolic measure for a C ² non-invertible but non-degenerate endomorphism on a compact Riemannian manifold without boundary. Based on this, we give a new formula of Lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms. Our results extend several well known theorems of Barreira et al. (Ann Math 149:755–783, 1999) and Ledrappier and Young [Commun Math Phys 117(4):529–548, 1988] for diffeomorphisms to the case of endomorphisms.