The dynamical properties of Penrose tilings

The Dynamical Properties of Penrose Tilings

The set of Penrose tilings, when provided with a natural compact metric topology, becomes a strictly ergodic dynamical system under the action of R2 by translation. We show that this action is an almost 1:1 extension of a minimal R2 action by rotations on T4, i.e., it is an R2 generalization of a Sturmian dynamical system. We also show that the inflation mapping is an almost 1:1 extension of a ...

Penrose Tilings, Chaotic Dynamical Systems and Algebraic K-Theory

In this article we initiate the use of noncommutative geometry in the theory of dynamical systems. After investigating by examples the unusual and striking elementary properties of the Penrose tilings and the Arnold cat map, we associate a finite symbolic dynamics with finite grammar rules to each of them. Instead of studying these Markovian systems with the help of set-topology, which would gi...

Homogenization of Penrose tilings

where Ω is an open subset of R, and f depends on x also through the shape and the orientation of the cell contaning x in an a-periodic tiling of the space R. As an example we may consider mixtures of two (linear) conducting materials with different dielectric constants depending on the type of the tile, and f(y,Du) = a(y)|Du|, where a takes two values α, β depending whether y is in one type of ...

Deformed Penrose Tilings

Monte Carlo (MC) simulation of a model quasicrystal (2D Penrose rhomb tiling) shows that the kinds of local distortions that result from size-effect-like relaxations are in fact very similar to mathematical constructions called deformed model sets. Of particular interest is the fact that these deformed model sets are pure point-diffractive, i.e. they give diffraction patterns that have sharp Br...

Subdivision Surfaces and Penrose Tilings