Hausdorff Dimension Estimates for Invariant Sets of Time-Dependent Vector Fields

Abstract

In this paper we are concerned with generalizations of the results of A. Douady and J. Oesterlé [4] on estimates for the Hausdorff dimension of sets on Riemannian manifolds being negatively invariant with respect to a map. The main theorem that we derive for maps allows a number of corollaries which generalize several other results of A. V. Boichenko, F. Ledrappier and C. A. Leonov (see [2, 7, 8]). We extend the concept on differential equations and the corresponding vector fields on Riemannian manifolds. To obtain upper bounds for the Hausdorff dimension we formulate conditions for the eigenvalues of the symmetric part of the covariant derivative of the vector field. Modifications of the eigenvalues by the choice of an appropriate Riemannian metric will be of great importance. Besides the investigation of dimension of negatively invariant sets we are interested in the convergence behaviour of autonomous differential equations on Riemannian manifolds. We propose also a general form of the Bendixson-Dulac criterion for the non-existence of non-trivial periodic orbits of vector fields on compact Riemannian manifolds.