Fractal dimension estimate for invariant set in complete Riemannian manifold

In this paper, we give a simple upper bounds for the fractal dimensions of a forward invariant set and a negatively invariant set of a C1-diffeomorphism of the complete Riemannian manifold with non-negative Ricci curvature. These results generalize the corresponding result of C. Robinson [Dynamics systems: stability symbolic dynamics and chaos. 2nd ed. Boca Raton, London, New York, Washington: CRC Press; 1999] in RnRn, and weakens the condition to the singular values of the tangent map in [Boichenko VA, Franz A, Leonov GA, Reitmann V. Hausdorff and fractal dimension estimates for invariant sets of non-injective maps. Z Anal Anw 1998;17:207–23]. Finally, as application, we give the upper bound of the fractal dimension of an invariant set of the flow on Riemannian manifold.