TSpace

Abstract (summary): The thesis consists of two main results. The first main result is a proof that C^1 rigidity holds for circle maps with a break point for almost all rotation numbers. The second main result is a proof that C^1 robust rigidity holds for circle maps in the fractional linear transformation (FLT) pair family. That is, for this family, C^1 rigidity holds for all irrational rotation numbers. The approach taken here of proving a more general theorem that C^1 rigidity holds for circle maps with a break point satisfying a `derivatives close condition', allows us to obtain both of our main results as corollaries of this more general theorem.