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Hausdorff and packing measure functions of self-similar sets: continuity and measurability

Published online by Cambridge University Press:  01 October 2008

L. OLSEN
L. OLSEN*
Affiliation:
Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland, UK (email: [email protected])
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Abstract

Let N be an integer with N≥2 and let X be a compact subset of ℝd. If is a list of contracting similarities Si:X→X, then we will write for the self-similar set associated with , and we will write M for the family of all lists satisfying the strong separation condition. In this paper we show that the maps (1)and (2)are continuous; here denotes the Hausdorff dimension, ℋs denotes the s-dimensional Hausdorff measure and 𝒮s denotes the s-dimensional spherical Hausdorff measure. In fact, we prove a more general continuity result which, amongst other things, implies that the maps in (1) and (2) are continuous.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1635 - 1655
Copyright
Copyright © 2008 Cambridge University Press

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