Tilings, substitution systems and dynamical systems generated by them (Q1812692)

The tiling problem is the question whether, given a finite set of unit square domain tiles with certain adjacency rules (``tiling system''), there exists a tiling of the entire plane using copies of these tiles: this problem has been proved undecidable by Berger. The author of the paper under review associates to a tiling system a dynamical system consisting of all tilings of the planes using this tiling system with the natural action of \(\mathbb{Z}^ 2\) by translations. He addresses the question of checking what classes of dynamical systems can be realized this way. He studies in particular relations between \(2-D\) substitution systems and tiling systems from the point of view of dynamical systems, giving several nice examples. Finally he sketches a new proof of the undecidability of the tiling problem.