Fractal Geometry Seminar

“Self Affine Maps”

Many historical examples of fractals, such as the Cantor Set (Smith, 1874, Cantor 1883), Sierpin- ski’s Triangle (Cosmati 13th Century, Sierpinski, 1915) and the Koch curve (Koch 1904) are special cases of a more general construction, called Iterated Function Systems (IFS). Let f1,f2,...,fn be a set of contraction maps. We define the IFS based on f1, f2, . . . , fn as the unique non-trivial compact operator K such that K = ∪fi(K). In this talk we consider the very simple family of contraction maps, f1(x, y) = (μx−1, λx−1) and f2(x, y) = (μx+1, λx+1) where 0 ≤ λ, μ < 1. We will investigate how properties of this IFS vary as μ and λ vary, demonstrating a surprisingly rich structure.