Question

The Cantor set, C, is the set of real numbers r for which Tⁿ(r) ϵ [0,1] for all n, where T is the tent transformation. If we set C₀= [0,1], then we can recursively define a sequence of sets C_i, each of which is a union of 2ⁱ intervals of length 3^-i as follows: C_i+1 is obtained from C_i by removing the (open) middle third from each interval in C_i. We then can define the Cantor set by

C= i=0 to infinity C_i
In general, a set S is called self-similar if for some real number r the scale of S by r can be exactly covered (without overlap) by a finite number, say n, of copies of the original set S. Then if r^d=n we say that d is the similarity dimension of the set 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.

Answer 1

(a) The cantor set is obtained by successively removing intervals.

      We will measure the lenght of intervals removed.

      We know that at each step the number of intervals doubles and their length is decreased by 3.

      Therefore the length of intervals removed  

                                                                        

     Which is geometric series with common ratio , therefore it converges to .

     Because sum of geometric series whose common ratio is r and first term a is

     Therefore the length of interval removed

                                                                     

                                                                    

                                                                   

    So the length of interval removed = 1

    Length of cantor set = length of interval [0, 1] - length of interval removed

                                    = 1 - 1

                                    = 0

(b) Generalized Cantor set consist of two copies of itself, scaled by the factor x.

     Therefore it's similarity dimension

     Here scaled factor is 1/3 therefore the similarity dimension

.