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Abstract: For 0<λ<1 we consider the compact invariant set
("attractor") of the iterated function system F&lambda defined
by the three maps fi=λI + pi in
the plane, where p0=(0,0),
p1=(1-λ,0),
p2=(1-λ)(½,1). For &lambda=½ the
attractor is the Sierpinski triangle. For λ in the intervals [.6439,.6441],
[.6458,.6466], and [.6470,.6472] standard techniques for determining the Hausdorff
dimensio... read moren (or Lebesgue measure) of the "fat" Sierpinski triangle do not apply, and the
Hausdorff dimension has not been known for any specific such λ. For all these
λ we show that the attractor has nonempty interior and hence positive Lebesgue
measure and Hausdorff dimension 2. The novelty of our approach is that instead of extending
techniques developed for small λ we significantly extend geometric methods
developed by Broomhead, Montaldi and Sidorov for larger values of
λ.
Thesis (Ph.D.)--Tufts University, 2012.
Submitted to the Dept. of Mathematics.
Advisor: Boris Hasselblatt.
Committee: Zbigniew Nitecki, Fulton Gonzalez, and Robert
Devaney.
Keyword: Mathematics.read less
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