Distribution of the displacement sequence of an orientation preserving circle homeomorphism

作者：Marzantowicz Waclaw^*; Signerska Justyna

来源：Dynamical Systems, 2014, 29(2): 153-166.

DOI：10.1080/14689367.2013.868867

• circle homeomorphism
• displacement distribution
• rotation number
• integrate-and-fire

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 phi with irrational rotation number rho(phi). The values of displacement are dense in a set which depends on the map gamma (semi-)conjugating phi with the rotation by rho(phi) and which is the support of the displacement distribution. We provide an effective formula for the density of this distribution if gamma is a C-1-diffeomorphism. Moreover, we show approximation of the displacement distribution by sample displacements measured for any other circle homeomorphism sufficiently close to the initial homeomorphism phi, which constitutes a rigorous proof of numerical results seen in certain integrate-and-fire models.

• 出版日期2014-4-3