scholarly journals The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping

2002 ◽  
Vol 31 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Hongwen Guo ◽  
Dihe Hu
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Intersection Property ◽  
Random Sets ◽  
Hausdorff Measures ◽  
Open Set Condition ◽  
Finite Intersection ◽  
Open Set ◽  
Finite Intersection Property

We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.

HAUSDORFF MEASURES OF A CLASS OF MORAN SETS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050053
Author(s):  
XIAOFANG JIANG ◽  
QINGHUI LIU ◽  
GUIZHEN WANG ◽  
ZHIYING WEN
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Hausdorff Measures ◽  
Dimensional Hausdorff Measure ◽  
Self Similar

Let [Formula: see text] be the class of Moran sets with integer [Formula: see text] and real [Formula: see text] satisfying [Formula: see text]. It is well known that the Hausdorff dimension of any set in this class is [Formula: see text]. We show that for any [Formula: see text], [Formula: see text] where [Formula: see text] denotes [Formula: see text]-dimensional Hausdorff measure of [Formula: see text]. For any [Formula: see text] with [Formula: see text] there exists a self-similar set [Formula: see text] such that [Formula: see text].

On the connexion between Hausdorff measures and generalized capacity

1961 ◽  
Vol 57 (3) ◽  
pp. 524-531 ◽  
Author(s):  
S. J. Taylor
Keyword(s):  
Euclidean Space ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Real Function ◽  
Function Theory ◽  
Logarithmic Capacity ◽  
Hausdorff Measures ◽  
Positive Capacity ◽  
Special Case

For any real function h(t) which is continuous and monotonic increasing for t > 0 with , Hausdorff (10) in 1918 denned a Carathéodory measure with respect to h(t) which has subsequently been known as Hausdorff measure. For analysing sets in Euclidean space, these measures have proved both useful and interesting. Given a real function Φ(t) which is continuous and monotonic decreasing for t > 0 with , Frostman(9) in 1935 denned capacity with respect to Φ(t). Lebesgue measure in Euclidean k-space is a special case of Hausdorff measure, and capacity with respect to Φ(t) becomes logarithmic capacity or Newtonian capacity in the cases , Φ(t)=1/t, respectively. The interrelationship between h-measure and Φ-capacity has been of interest in both directions: (i) in applications to function theory one may be able to determine whether or not a set has positive capacity by examining the h-measure for suitable h(t) (see, for example, (5)); (ii) it may be possible to determine the measure properties of a set from knowledge of its capacity (see, for example, (7) and (17)).

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