In the field of dynamical systems, it is customary to oppose ordered dynamical systems, and chaotic dynamical systems. This opposition can be expressed in several different ways: systems of entropy zero versus systems of strictly positive entropy, systems with low (polynomial) complexity versus systems with exponential complexity, systems without or with sensitive dependence to initial conditions... . In this paper, we would like to show, in a specific elementary case, that there can be a remarkable relation between these two kinds of systems; by considering a collection of ordered systems, and a renormalization operation on these systems, we can observe chaotic dynamics. The relation between these two dynamics can be expressed in a variety of ways, geometric, symbolic or arithmetic, linking well-known mathematical theories. We will study the simplest nontrivial quasicrystals: the one-dimensional quasicrystals obtained by the "cut and project" method in the plane, the best known being the so-called Fibonacci quasicrystal, a one-dimensional analogue of the Penrose tiling.
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