Hausdorff dimension estimates for invariant sets of time-dependent vector fields (Q1915227)

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).