Irregularity Index and Spherical Densities of the Penta-Sierpinski Gasket
We compute the centred Hausdorff measure, C s ( P ) ∼ 2.44 , and the packing measure, P s ( P ) ∼ 6.77 , of the penta-Sierpinski gasket, P , with explicit error bounds. We also compute the full spectra of asymptotic spherical densities of these measures in P , which, in contrast with that of the Sie...
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| Veröffentlicht in: | Mediterranean journal of mathematics 2023-12, Vol.20 (6), Article 322 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | We compute the centred Hausdorff measure,
C
s
(
P
)
∼
2.44
, and the packing measure,
P
s
(
P
)
∼
6.77
, of the penta-Sierpinski gasket,
P
, with explicit error bounds. We also compute the full spectra of asymptotic spherical densities of these measures in
P
, which, in contrast with that of the Sierpinski gasket, consists of a unique interval. These results allow us to compute the irregularity index of
P
,
I
(
P
)
∼
0.6398
, which we define for any self-similar set
E
with open set condition as
I
(
E
)
=
1
-
C
s
(
E
)
P
s
(
E
)
. |
|---|---|
| ISSN: | 1660-5446 1660-5454 |
| DOI: | 10.1007/s00009-023-02528-6 |