Upper bounds for the Hausdorff dimension of compact invariant sets of \(C^1\)-maps on smooth Riemannian manifolds are given in terms of the singular values of the tangent map and the multiplicity function of the map.
The singular values of the linear operator \(L\) are defined as the eigenvalues of the operator \(\sqrt{L^*L}\). The main idea consists in showing that the map is contracting with respect to Hausdorff outer measure on the invariant set. The contraction constant can be estimates by means of the singular values of the tangent map.
The multiplicity function of the map is defined as the cardinality of the pre-images of the points. It is shown how the multiplicity function of a map can be used to prove that the map is contracting with respect to the weighted Hausdorff outer measure.
The obtained estimates can be applied to a wide class of non-invertible maps and to iterated function systems. To demonstrate the strength of the presented method some simple examples for which the obtained estimates can be compared with the exact value or to known estimates are considered.