Iterated Function Systems of Interval Maps
M. Gharaei
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
P
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.
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