Self-similar sets in complete metric spaces

Author:
Andreas Schief

Journal:
Proc. Amer. Math. Soc. **124** (1996), 481-490

MSC (1991):
Primary 28A80, 28A78

MathSciNet review:
1301047

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Abstract | References | Similar Articles | Additional Information

Abstract: We develop a theory for Hausdorff dimension and measure of self-similar sets in complete metric spaces. This theory differs significantly from the well-known one for Euclidean spaces. The open set condition no longer implies equality of Hausdorff and similarity dimension of self-similar sets and that has nonzero Hausdorff measure in this dimension. We investigate the relationship between such properties in the general case.

**1**Christoph Bandt and Siegfried Graf,*Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure*, Proc. Amer. Math. Soc.**114**(1992), no. 4, 995–1001. MR**1100644**, 10.1090/S0002-9939-1992-1100644-3**2**K. J. Falconer,*The geometry of fractal sets*, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR**867284****3**N. Martinov and N. Vitanov,*On the correspondence between the self-consistent 2D Poisson-Boltzmann structures and the sine-Gordon waves*, J. Phys. A**25**(1992), no. 2, L51–L56. MR**1147131****4**J. E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), 713--747.**5**D. G. Larman,*A new theory of dimension*, Proc. London Math. Soc. (3)**17**(1967), 178–192. MR**0203691****6**Andreas Schief,*Separation properties for self-similar sets*, Proc. Amer. Math. Soc.**122**(1994), no. 1, 111–115. MR**1191872**, 10.1090/S0002-9939-1994-1191872-1**7**Sergio Stella,*On Hausdorff dimension of recurrent net fractals*, Proc. Amer. Math. Soc.**116**(1992), no. 2, 389–400. MR**1094507**, 10.1090/S0002-9939-1992-1094507-X

Sergio Stella,*Corrigendum to: “On Hausdorff dimension of recurrent net fractals” [Proc. Amer. Math. Soc. 116 (1992), no. 2, 389–400; MR1094507 (92m:58079)]*, Proc. Amer. Math. Soc.**121**(1994), no. 4, 1309–1311. MR**1224620**, 10.1090/S0002-9939-1994-1224620-7

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Additional Information

**Andreas Schief**

Affiliation:
Corporate Research and Development, SIEMENS AG, 81730, Munich, Germany

Email:
andreas.schief@zfe.siemens.de

DOI:
https://doi.org/10.1090/S0002-9939-96-03158-9

Keywords:
SOSC,
OSC,
self-similar sets,
fractals,
Hausdorff dimension,
complete metric spaces

Received by editor(s):
June 9, 1994

Received by editor(s) in revised form:
August 23, 1994

Communicated by:
Christopher D. Sogge

Article copyright:
© Copyright 1996
American Mathematical Society