Relations Between Entropies,Exponents And Dimensions

We focus on the relations between entropies,exponents and dimensions for differentiable dynamics.We propose a simplified proof of Ledrappier-Young en-tropy formula.Then by using Ledrappier-Young entropy formula and Pesin the-ory,we bound the entropy of an ergodic measure by a combination of Lyapunov exponents and various invariants on the unstable manifolds such as volume growth,topological entropies and fibered entropies.We also systemically investigate the so called dimensional entropies and their applications to entropy-hyperbolic systems.