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C. f-+ L~= l d n 3- n is a homeomorphism From Fractal Groups to Fractal Sets 35 The standard iterated function system on {0,2}W can be identified via the map with the iterated function system on C consisting of two maps ¢o (x) = x /3 and ¢1(X) = x/3 + 2/3. It is easy to see now that the triadic numeration system gives exactly the standard encoding of the Cantor set with respect to the described iterated function system.

The transformation a generates an infinite cyclic group of transformations of the space {O, l}w. Thus we get an action of the group Z , which will be also called adding machine action. It is easy to see that the adding machine action is self-similar. 41 From Fractal Groups to Fractal Sets The dihedral group. Let a and b be the transformations of the space XW = {O, 1}W, defined by the rules (Ow)a (1w)a = = 1w Ow (OW)b (1w)b = Owa = 1w b, were w E XW is arbitrary. The group generated by the transformations a and b is isomorphic to the infinite dihedral group ITlloo, and thus we get a self-similar action of this group on XW.

At the moment very precise estimates of h exists, and it can also be computed with arbitrary precision (see [Boy73a, Boy73b] and [McM98]). C. T. 305688. The Apollonian gasket can be obtained by the same procedure of removing inscribed disks but starting from a curvilinear triangle formed by three tangent circles. It is a self-similar set with self-similarity structure defined by the rational transformations fi ' i = 1, 2,3, which map the triangle onto its three subtriangles. For instance, if we take the triangle with the vertices 1, exp(27ri/3), exp( -27ri/3) then the iterated function system will be h(z) = (~,,;:-~z:ll, 12(z) = expe;i)h(z) and 13(z) = exp( -27rij:3)h (z) (see [MU98]).