Self-similar sets of zero Hausdorff measure and positive packing measure

2000 ◽  
Vol 117 (1) ◽  
pp. 353-379 ◽  
Author(s):  
Yuval Peres ◽  
Károly Simon ◽  
Boris Solomyak
Keyword(s):  
Hausdorff Measure ◽  
Packing Measure ◽  
Self Similar

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

2008 ◽  
Vol 28 (5) ◽  
pp. 1635-1655 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Hausdorff Measure ◽  
The Self ◽  
Packing Measure ◽  
Dimensional Hausdorff Measure ◽  
Continuity Result ◽  
The Family ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Image Position

AbstractLetNbe an integer withN≥2 and letXbe a compact subset of ℝd. If$\mathsf {S}=(S_{1},\ldots ,S_{N})$is a list of contracting similaritiesSi:X→X, then we will write$K_{\mathsf {S}}$for the self-similar set associated with$\mathsf {S}$, and we will writeMfor the family of all lists$\mathsf {S}$satisfying the strong separation condition. In this paper we show that the maps(1)and(2)are continuous; here$\dim _{\mathsf {H}}$denotes the Hausdorff dimension, ℋsdenotes thes-dimensional Hausdorff measure and 𝒮sdenotes thes-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.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

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].

Geometric measures for parabolic rational maps

1992 ◽  
Vol 12 (1) ◽  
pp. 53-66 ◽  
Author(s):  
M. Denker ◽  
M. Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Rational Maps ◽  
Packing Measure ◽  
Rational Map ◽  
Dimensional Hausdorff Measure ◽  
Conformal Measure

AbstractLet h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

scholarly journals On the packing measure of slices of self-similar sets

2015 ◽  
Vol 2 (4) ◽  
pp. 389-401 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Packing Measure ◽  
Self Similar
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