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We investigate the regularity properties of the displacement sequence  where is a lift of an orientation preserving circle homeomorphism. If the rotation number  is rational, then  is asymptotically periodic with semi-period q. This convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. This leads to the notion of an  edge, which we illustrate by the numerical example. If  then some classical results in topological dynamics yield that the displacement sequence also exhibits some regularity properties, which we define and prove in the second part of the paper. 

circle homeomorphism, displacement sequence, integrate-and-fire models.