An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks
Akhtam DzhalilovTurin Polytechnic University Kichik Halka yuli 17, Tashkent, 100095 UzbekistanDaniel MayerNational University of Uzbekistan VUZ Gorodok, Tashkent, 700174 UzbekistanInstitut f ̈ ur Theoretische PhysikSamarkand Institute of Economics and Service A. Temura st. 9, Samarkand, 140100 UzbekistanS. DjalilovTurin Polytechnic University Kichik Halka yuli 17, Tashkent, 100095 UzbekistanA. Aliyev
ABI
Abstract
showed that the invariant measure h of a piecewise linear (PL) circle homeomorphism h with two break points and an irrational rotation number h is absolutely continuous iff the two break points belong to the same orbit. We extend Herman's result to the class P of piecewise C 2+ -circle maps f with an irrational rotation number f and two break points a 0 , c 0 , which do not lie on the same orbit and whose total jump ratio is f = 1, as follows: if f denotes the invariant measure of the P -homeomorphism f , then for Lebesgue almost all values of f ([a 0 , c 0 ]) the measure f is singular with respect to Lebesgue measure.