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We analyze the two-point motions of iterated function systems on the unitinterval generated by expanding and contracting affine maps, where theexpansion and contraction rates are determined by a pair $(M,N)$ of integers.  This dynamics depends on the Lyapunov exponent. For a negative Lyapunovexponent we establish synchronization, meaning convergence of orbits withdifferent initial points. For a vanishing Lyapunov exponent we establishintermittency, where orbits are close for a set of iterates of full density,but are intermittently apart. For a positive Lyapunov exponent we show theexistence of an absolutely continuous stationary measure for the two-pointdynamics and discuss its consequences.  For nonnegative Lyapunov exponent and pairs $(M,N)$ that are multiplicativelydependent integers, we provide explicit expressions for absolutely continuousstationary measures of the two-point motions. These stationary measures areinfinite $\sigma$-finite measures in the case of zero Lyapunov exponent. Forvarying Lyapunov exponent we find here a phase transition for the system oftwo-point motions, in which the support of the stationary measure explodes withintermittent dynamics and an infinite stationary measure at the transitionpoint.

Iterated function systems of affine expanding and contracting maps on the unit interval