For a C^1 map φ on a Riemannian manifold and for a compact invariant set K it is proven that the maximal local Lyapunov dimension of φ 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.
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Karin Gelfert, Maximum Local Lyapunov Dimension Bounds the Box Dimension. Direct Proof for Invariant Sets on Riemannian Manifolds. Z. Anal. Anwend. 22 (2003), no. 3, pp. 553–568DOI 10.4171/ZAA/1162