Small islands of stability in the chaotic sea

In the present work we investigate the model Carleson map on two dimensional torus. The Carleson map is a typical example of an area-preserving map. We consider the problem of finding of the some types of islands of stability for Carleson map. In this work we describe the islands of stability of period 1, period 2 and particular cases of the islands of stability of period 3 and period 4. The purpose of the present work is to find such areas in the phase space and intervals for real parameter g for Carleson map, when the islands of stability exist. The complication of the problem is that the measure of the islands of stability is less than the measure of the surrounding chaotic area