Abstract

For a special class of non–injective maps on Riemannian manifolds upper and lower bounds for the Hausdorff dimension of invariant sets are given in terms of the singular values of the tangent map. The upper estimation is based on a theorem by Douady and Oesterlé and its generalization to Riemannian manifolds by Noack and Reitmann, but additionally information about the noninjectivity is used. The lower estimation can be reached by modifying a method, derived by Shereshevskij for geometric constructions on the real line (also described by Barreira, for similar constructions in general metric spaces. The upper and lower dimension estimates for k — 1 — endomorphisms can for instance be applied to Julia sets of quadratic maps on the complex plane.

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