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

scholarly journals Computability of the packing measure of totally disconnected self-similar sets

2015 ◽  
Vol 36 (5) ◽  
pp. 1534-1556 ◽  
Author(s):  
MARTA LLORENTE ◽  
MANUEL MORÁN
Keyword(s):  
Sierpinski Gasket ◽  
Separation Condition ◽  
Packing Measure ◽  
Sierpiński Gasket ◽  
Additional Information ◽  
Strong Separation ◽  
Self Similar ◽  
Contraction Factor ◽  
Strong Separation Condition ◽  
Totally Disconnected

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called Strong Separation Condition (SSC) and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contraction factor in the interval $(0,\frac{1}{3}]$.

scholarly journals On the packing measure of self-similar sets

Nonlinearity ◽  
2013 ◽  
Vol 26 (11) ◽  
pp. 2929-2934 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Packing Measure ◽  
Self Similar

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

2011 ◽  
Vol 32 (3) ◽  
pp. 1101-1115 ◽  
Author(s):  
HUA QIU
Keyword(s):  
Iterated Function Systems ◽  
Packing Measure ◽  
Open Set Condition ◽  
Open Set ◽  
Density Theorems ◽  
Iterated Function ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Image Position

AbstractIn 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 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].

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.

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

Self-similar collapse with non-radial motions

2006 ◽  
Vol 20 ◽  
pp. 1-4
Author(s):  
A. Nusser
Keyword(s):  
Self Similar
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