Random Substitutions and Fractal Percolation.

摘要

Although we shall touch on quite complicated phenomena in this thesis, like turbulence in a fluid and the propaganda of gene information through generations, we start from a simple mathematical object: that of substitutions on words. A word is a sequence of symbols, for instance, 0's and 1's. An example of a substitution on words is the Fibonacci substitution that replaces 0's by 01 and 1's by 0. If we apply this substitution on the word 0101 we obtain the word 010010.. We get a sequence of words by repeatedly applying the substitution on the outcome of the previous substitution. Starting with a 0, we obtain the sequence 0, 01, 010, 01001, 01001010, 0100101001001 and so on. The lengths of these words, i.e., The lengths of these words, i.e., 1, 2, 3, 5, 8, 13,, yield the famous Fibonacci numbers. Substitutions can also be applied to sets in the plane.