The Cantor set, C, is the set of real numbers r for whichTn(r) ? [0,1] for all n, where T is the tenttransformation. If we set C0= [0,1], then we canrecursively define a sequence of sets Ci, each of whichis a union of 2i intervals of length 3-i asfollows: Ci+1 is obtained from Ci by removingthe (open) middle third from each interval in Ci. Wethen can define the Cantor set by
C= i=0 to infinityCi
In general, a set S is called self-similar if for some real numberr the scale of S by r can be exactly covered (without overlap) by afinite number, say n, of copies of the original set S. Then ifrd=n we say that d is the similarity dimension of theset S.
1. Consider the Cantor set as described above.
a. What is the length of the Cantor set?
b. Find the similarity dimension of the Cantor set.
