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Continuity of packing measure functions of self-similar iterated function systems

Published online by Cambridge University Press:  24 May 2011

HUA QIU
HUA QIU*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, China (email: huatony@gmail.com)
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Abstract

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In this paper, we focus on the packing measures of self-similar sets. Let K be a self-similar set whose Hausdorff dimension and packing dimension equal s. We state that if K satisfies the strong open set condition with an open set 𝒪, then \[ \mathcal {P}^s(K\cap B(x,r))\geq (2r)^s \] for each open ball B(x,r)⊂𝒪 centred in K, where 𝒫s denotes the s-dimensional packing measure. We use this inequality to obtain some precise density theorems for the packing measures of self-similar sets. These theorems can be used to compute the exact value of the s-dimensional packing measure 𝒫s (K)of K. Moreover, by using the above results, we show the continuity of the packing measure function of the attractors on the space of self-similar iterated function systems satisfying the strong separation condition. This result gives a complete answer to a question posed by Olsen in [15].

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012, pp. 1101 - 1115
Copyright
Copyright © Cambridge University Press 2011

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