Dynamics of Group Actions seminar
Consider a $C^2$ non-invertible but non-degenerate endomorphism $f$ on a compact Riemannian manifold without boundary. We are interested in the dimension theories of $f$-invariant measures. By using the notion of folding entropy introduced by Ruelle, we set up an equality relating entropy, folding entropy, dimension, and negative Lyapunov exponents. Based on this, we establish the exact dimensional property of an ergodic hyperbolic measure. We also give a new formula of Lyapunov dimension for ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms. An application of folding entropy to the theory of entropy production is also addressed. Our results extend several well known results of Ledrappier-Young and Barreira et al. for diffeomorphisms to the case of endomorphisms.