4 Dec
2005
4 Dec
'05
11:41 p.m.
Kerry writes: << . . . [several interesting examples of "fractal" sequences, and] . . . Since the exact manner in which a sequence can contain copies of itself varies, I don't know that there is an exact definition of a fractal sequence.
Hmm. Suppose we take a random sequence of 0's and 1's. With probability 1, every initial string of this sequence is later repeated, nay, repeated infinitely often. Does that mean that except for a set of measure zero (in the product space {0,1}^N endowed with the product measure {1/2,1/2}^N), every sequence is fractal? (N is your favorite definition of the natural numbers.) --Dan