ATTRACTION DOMAIN FOR THE ATTRACTOR OF A TWO-COLOR CIRCLE ROTATION

摘要

The dynamics of contraction of the unit circle C under iterated two-color circle rotation S_εe, dependent on a continuous parameter ε ∈ C, is studied. The dynamic distance from the deep hole Dh(ε) of the attraction domain Spir_ε = C Att_ε to the attractor Att_ε of the rotation S_ε and the measure of the deep hole |Dh(ε)| are computed. It is proved that as e ↑ 1, the phenomenon of localization of the deep holes Dh(ε) occurs. It is shown that the process of contraction of the circle, C is contained in S_ε(C) is contained in S_ε~2(C) is contained in …is contained in S_ε~k(C) is contained in ···, goes on in three linear modes if the parameter e coincides with an eigenvalue ε_m of a certain B-process, and in four modes in the general case.