On the boundary of attractors

with non-void interior

Authors:
Ka-Sing Lau and You Xu

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1761-1768

MSC (2000):
Primary 28A80, 52C22; Secondary 28A78

DOI:
https://doi.org/10.1090/S0002-9939-99-05303-4

Published electronically:
October 27, 1999

MathSciNet review:
1662265

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a family of contractive mappings on such that the attractor has nonvoid interior. We show that if the 's are injective, have non-vanishing Jacobian on , and have zero Lebesgue measure for then the boundary of has measure zero. In addition if the 's are affine maps, then the conclusion can be strengthened to . These improve a result of Lagarias and Wang on self-affine tiles.

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Additional Information

**Ka-Sing Lau**

Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

Email:
kslau@math.cuhk.edu.hk

**You Xu**

Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Email:
yoxst+@pitt.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05303-4

Keywords:
Boundary,
Hausdorff dimension,
self-affine tiles,
self-similarity,
singular values

Received by editor(s):
January 8, 1998

Received by editor(s) in revised form:
July 23, 1998

Published electronically:
October 27, 1999

Additional Notes:
The first author was partially supported by the RGC grant CUHK4057/98P

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society