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Let f be a C1 self-map on a smooth Riemannian manifold M , μ be an f -invariant ergodic Borel probability measure with a compact support Λ and χμ > · · · > χμ be the Lyapunov exponents of μ with respect to f . If χμ > 0, then we give a lower bound of the lower pointwise dimension of μ in terms of χμ and of the entropy hμ(f). Moreover, if Df{·} is non-degenerate on Λ and χμ > 0, then we give an upper bound of the upper pointwise dimension of μ in terms of χμ and of the entropy hμ(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 hμ(f).

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