Unfolding numbers for interval maps

We propose a new invariant (rotation number) for a cycle of an interval map, called its unfolding number to circumvent some difficulties faced in the over-rotation theory for interval maps. We corroborate the completeness of the theory and exhibit that the set of unfolding numbers of points under a given interval map f forms an interval of the form [uf, 1/2] where uf can be computed by a simple algorithm for any interval map f. Further, uf is realized on a closed invariant subset Zf ⊂ [0,1] such that f|Zf is strictly ergodic. In the end we forge connection between unfolding numbers and over-rotation numbers: we reveal that while disparate in essence, there exists special patterns πₛ called sheer patterns for whom unfolding numbers equals over-rotation numbers.