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

Advances in Mathematics, vol. 482, pp. 110605

Abstract

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. 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.

Authors 2

1. Ale Jan Homburg Aachen Mathematics Institute

   Leiden University · University of Amsterdam

   Affiliation as printed

   KdV Institute for Mathematics, University of Amsterdam, Science Park 107, 1098 XG Amsterdam, Netherlands

   Mathematics Institute, Leiden University, Einsteinweg 55, 2333 CC Leiden, Netherlands

2. Charlene Kalle corresponding Aachen Mathematics Institute

   Leiden University

   Affiliation as printed

   Mathematics Institute, Leiden University, Einsteinweg 55, 2333 CC Leiden, Netherlands