Hausdorff dimension of deranged Cantor set without some boundedness condition

In-Soo Baek

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2004; 19(1): 113-117

Printed March 1, 2004

Hausdorff dimension of deranged Cantor set without some boundedness condition

In-Soo Baek

Pusan University of Foreign Studies

Abstract

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Abstract

A deranged Cantor set(without the uniform boundedness condition away from zero of contraction ratios) whose weak local dimensions for all points coincide has its Hausdorff dimension of the same value of weak local dimension. We will show it using an energy theory instead of Frostman's density lemma which was used for the case of the deranged Cantor set with the uniform boundedness condition of contraction ratios. In the end, we will give an example of such a deranged Cantor set.

Keywords: Hausdorff dimension, Cantor set, weak local dimension

MSC numbers: 28A78