RECODING STURMIAN SEQUENCES ON A SUBSHIFT OF FINITE TYPE CHAOS FROM ORDER: A WORKED OUT EXAMPLE

摘要

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.