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

We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers. This dynamics depends on the Lyapunov exponent. For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existence of an absolutely continuous stationary measure for the two-point dynamics and discuss its consequences. For nonnegative Lyapunov exponent and pairs $(M,N)$ that are multiplicatively dependent integers, we provide explicit expressions for absolutely continuous stationary measures of the two-point motions. These stationary measures are infinite $σ$-finite measures in the case of zero Lyapunov exponent. For varying Lyapunov exponent we find here a phase transition for the system of two-point motions, in which the support of the stationary measure explodes with intermittent dynamics and an infinite stationary measure at the transition point.