Iterated function systems of interval maps

This thesis contains three related articles which are inserted in separate chapters. The chapters are self contained, hence it is possible to read each of them independently. In each of the chapters we discuss iterated function systems, IFSs, generated by finitely many continuous maps f1, . . . , fk on the unit interval I = [0, 1], taken with positive probabilities pi, ∑ki=1 pi =1, randomly at each iterate. We always assume that these maps fix the boundaries of the interval I and study dynamics of the system under some conditions like the sign of the Lyapunov exponent at the boundaries of the interval. The sign of Lyapunov exponent at a boundary determines whether the boundary is, on average, attracting (negative Lyapunov exponent), neutral (zero), or repelling (positive). We discuss occurrence of intriguing dynamical phenomena such as synchronization, intermingled basins and intermittency, in this context