Space tilings and substitutions

We generalize the study of symbolic dynamical systems of finite type and ℤ2 action, and the associated use of symbolic substitution dynamical systems, to dynamical systems with ℝ2 action. The new systems are associated with tilings of the plane. We generalize the classical technique of the matrix of a substitution to include the geometrical information needed to study tilings, and we utilize rotation invariance to eliminate discrete spectrum. As an example we prove that the pinwheel tilings have no discrete spectrum.