Distribution of the displacement sequence of an orientation preserving circle homeomorphism

Abstract

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 ϕ with irrational rotation number ϱ(ϕ). The values of displacement are dense in a set which depends on the map γ (semi-)conjugating ϕ with the rotation by ϱ(ϕ) 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 ϕ, which constitutes a rigorous proof of numerical results seen in certain integrate-and-fire models.