For the three vertices of an equilateral triangle,
the maps have the Sierpiński
sieve as their attractor. The illustration shows
the three images of the initial triangle and a seventh-stage
approximation, colored by the first map. The golden
gaskets use the same three centers with specially chosen contraction ratios greater
than ,
producing overlapping images (Broomhead et al. 2004).
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M. F. "Fractal Image Compression." Not. Amer. Math. Soc.43,
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Function Systems and the Global Construction of Fractals." Proc. Roy. Soc.
London, Ser. A399, 243-275, 1985.Barnsley, M. F. and
Hurd, L. P. Fractal Image Compression. Wellesley, MA: A K Peters, 1993.Bogomolny,
A. "The Collage Theorem." https://www.cut-the-knot.org/ctk/ifs.shtml.Broomhead,
D.; Montaldi, J.; and Sidorov, N. "Golden Gaskets: Variations on the Sierpiński
Sieve." Nonlinearity17, 1455-1480, 2004. https://doi.org/10.1088/0951-7715/17/4/017.Diaconis,
P. M. and Shashahani, M. "Products of Random Matrices and Computer Image
Generation." Contemp. Math.50, 173-182, 1986.Fisher,
Y. Fractal
Image Compression. New York: Springer-Verlag, 1995.Hutchinson,
J. E. "Fractals and Self-Similarity." Indiana Univ. Math. J.30,
713-747, 1981. https://doi.org/10.1512/iumj.1981.30.30055.Trott,
M. The
Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. §1.1.1.
https://www.mathematicaguidebooks.org/.Wagon,
S. "Iterated Function Systems." §5.2 in Mathematica
in Action. New York: W. H. Freeman, pp. 149-156, 1991.
for 
,
2, ..., 
,
where 
,
each with a contractivity factor 
, mapping a nonempty compact metric space
into itself. It is the basis for fractal image compression
techniques.
, called its attractor, satisfying
in the Hausdorff metric
(Hutchinson 1981).
of an equilateral triangle,
the maps 
have the Sierpiński
sieve as their attractor. The illustration shows
the three images of the initial triangle and a seventh-stage
approximation, colored by the first map. The golden
gaskets use the same three centers with specially chosen contraction ratios greater
than 
,
producing overlapping images (Broomhead et al. 2004).
independently at each step and setting 
. For positive probabilities 
with 
, the stationary probability
measure satisfies
is a Borel set and 
denotes its pre-image.
Changing these probabilities can change the measure
while leaving its support 
unchanged (Hutchinson 1981). When the maps
are similarities, this is a self-similar
measure.
