I have been experimenting with LZ77 (naively $O(n^2)$ runtime, infinite window). Applying it to the 7th Fibonacci word $abaababaabaab$ yields the correct LZ factorization:
My question is about the behavior of LZ77 if we iterate it. My experiments suggest that reapplication of LZ77 to the input will yield no further patterns that were not found the first time.
By reapplication I mean, where in the first instance we treat the factors of the string as the sequence of unit symbols 'a' and 'b', in the second application the factors are the LZ factors. I was hoping to discover (over larger various texts, like the Complete Sonnets of Shakespeare) increasing gains, and possibly, "multilevel" patterns found by LZ over the sequence of factors of the previous iterate. But none of this occurred. The sequence of factors after the second iteration is exactly the same as the first.
So where is the bug in my thinking? Is there a simple proof of this given the definition of an LZ factor being the longest prefix from the current position occurring in the concatenation of the preceding LZ factors?