Maximum local Lyapunov dimension bounds the box dimension. Direct proof for invariant sets on Riemannian manifolds (Q1431140)

Summary: For a \(C^1\) map \(\phi\) on a Riemannian manifold and for a compact invariant set \(K\) it is proved that the maximal local Lyapunov dimension of \(\phi\) on \(K\) bounds the box dimension of \(K\) from above. A version for Hilbert spaces is also presented. The introduction of an adapted Riemannian metric provides in a certain sense an optimal upper bound for the box dimension of the Lorenz attractor.