I am not sure yet about what I exactly need to prove, but I guess I can formulate a rough statement similar to the following:
Suppose $w\in F_2$ is a primitive word whose length is big enough. Then for every chunk of length greater than some big constant, there exists a "long" subword, contained in that chunk, which is primitive.
I suspect that the statement should be true, maybe in some similar form, and I thought about proving it by using some canonical form for primitive elements. Since the rank of the free group is just 2, I guess there should be an easy way for finding some good pattern in the primitive words, or something like that. Do you know any nice way of writing down primitive elements in $F_2$ that might be good in this context?
Thank you very much.