Hausdorff dimension estimates for invariant sets of time-dependent vector fields
type: Item
instance of: scholarly article
title: Hausdorff dimension estimates for invariant sets of time-dependent vector fields
zbMATH Open document ID: 0923.58029
DOI: 10.4171/ZAA/710
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-10-19T00:00:00Z
review text: Estimations of Hausdorff dimension of negatively invariants sets of time-dependent vector fields on compact Riemannian manifold are given. The Hausdorff dimension is a useful concept to study orbits of dynamical systems and their stabilities etc. Let \((M,g)\) be a compact Riemannian manifold without boundary with Betti number \(b_1\). Let \(f: M\to TM\) be a vector field of class \(C^{1}\) and \(u = f(u)\) be the corresponding differential equation. Then it is shown that under some conditions for \(\nabla f\), the above system has at most \(b_{1}\) non-trivial periodic orbits (Theorem 4.1).
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 35-XX
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 37N99
Mathematics Subject Classification ID: 58C25
zbMATH DE Number: 889007
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: dynamical system
zbMATH Keywords: stability
zbMATH Keywords: negatively invariant sets
zbMATH Keywords: time-dependent vector fields
zbMATH Keywords: Betti number
zbMATH Keywords: nontrivial periodic orbits
author: A. Noack
author: Volker Reitmann
reviewed by: Yasuhiro Asoo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 889007