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Zeitschrift für Analysis und ihre Anwendungen


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Volume 17, Issue 1, 1998, pp. 207–223
DOI: 10.4171/ZAA/816

Published online: 1998-03-31

Hausdorff and Fractal Dimension Estimates for Invariant Sets of Non-Injective Maps

V.A. Boichenko[1], A. Franz[2], G.A. Leonov[3] and Volker Reitmann[4]

(1) St. Petersburg State University, Russian Federation
(2) Technische Universität Dresden, Germany
(3) St. Petersburg State University, Russian Federation
(4) Technische Universität Dresden, Germany

In this paper we are concerned with upper bounds for the Hausdorff and fractal dimensions of negatively invariant sets of maps on Riemannian manifolds. We consider a special class of non-injective maps, for which we introduce a factor describing the "degree of non-injectivity". This factor can be included in the Hausdorif dimension estimates of Douady-Oesterlé type [2, 7, 10] and in fractal dimension estimates [5, 13, 15] in order to weaken the condition to the singular values of the tangent map. In a number of cases we get better upper dimension estimates.

Keywords: Hausdorff dimension estimates, fractal dimension estimates, non-injective maps, tangent map, singular values

Boichenko V.A., Franz A., Leonov G.A., Reitmann Volker: Hausdorff and Fractal Dimension Estimates for Invariant Sets of Non-Injective Maps. Z. Anal. Anwend. 17 (1998), 207-223. doi: 10.4171/ZAA/816