Zeitschrift für Analysis und ihre Anwendungen
Full-Text PDF (1122 KB) | Metadata | Table of Contents | ZAA summary
Published online: 1996-06-30
Hausdorif Dimension Estimates for Invariant Sets of Time-Dependent Vector FieldsA. Noack and Volker Reitmann (1) Technische Universität Dresden, Germany
(2) Technische Universität Dresden, Germany
In this paper we are concerned with generalizations of the results of A. Douady and J. Oesterlé  on estimates for the Hausdorif 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 apropriate 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.
Keywords: Riemannian manifolds, Hattsdorff dimension estimates, dynamical systems, global convergence, Bendizson-Dulac criterion
Noack A., Reitmann Volker: Hausdorif Dimension Estimates for Invariant Sets of Time-Dependent Vector Fields. Z. Anal. Anwend. 15 (1996), 457-473. doi: 10.4171/ZAA/710