Pointwise dimension, entropy and Lyapunov exponents for $C^{1}$ maps

Abstract

Let f be a C 1 self-map on a smooth Riemannian manifold M, μ be an f-invariant ergodic Borel probability measure with a compact support Λ and χ 1 μ > · · · > χs μ be the Lyapunov exponents of μ with respect to f. If χ 1 μ > 0, then we give a lower bound of the lower pointwise dimension of μ in terms of χ1 μ and of the entropy h μ (f). Moreover, if Df{·} is non-degenerate on Λ and χ s μ > 0, then we give an upper bound of the upper pointwise dimension of μ in terms of χs μ and of the entropy h μ (f). Furthermore, if f is C 1+α for some α > 0, then the non-degeneracy condition can be removed. As direct applications of the above results, we also give the corresponding lower and upper bounds of some classical characteristics of dimensional type of μ in terms of the Lyapunov exponents and of the entropy h μ (f). © 2012 American Mathematical Society.