Distribution of the displacement sequence of an orientation preserving circle homeomorphism

In some applications not only the knowledge of the behaviour of trajectories of a map is important, but also their displacements. We describe in detail the distribution of elements of the displacement sequence along a trajectory of an orientation preserving circle homeomorphism ? with irrational rotation number ?(?). The values of displacement are dense in a set which depends on the map γ (semi-)conjugating ? with the rotation by ?(?) and which is the support of the displacement distribution. We provide an effective formula for the density of this distribution if γ is a C¹-diffeomorphism. Moreover, we show approximation of the displacement distribution by sample displacements measured for any other circle homeomorphism sufficiently close to the initial homeomorphism ?, which constitutes a rigorous proof of numerical results seen in certain integrate-and-fire models.