Directional Expansiveness

Abstract

Expansiveness is an important property in dynamical systems related to symbolic dynamics and entropy. It was first studied in the context of continuous maps (Z-actions) and later generalized to continuous flows (R-actions). In the Z^d case, Boyle and Lind studied directional expansiveness. This followed Milnor's concept, directional properties, with the introduction of directional entropy. In the first part of the dissertation, we study expansive Z^d-actions, proving a Z^d version of the Keynes and Robertson theorem. We also find expansive directions for symbolic tensor product actions. In the second part of the dissertation, we develop a general theory of expansive directions for R^d-flows. This includes a definition of weak and strong directional expansiveness. Strong directional expansiveness implies weak directional expansiveness. The converse is false in general but holds for finite local complexity R^d-tiling dynamical systems. Our main result of the dissertation shows that the Penrose tiling dynamical system has exactly five non-strongly expansive directions. These are perpendicular to the 5^th roots of unity. The proof involves using Wang tiles to show that Penrose tilings are essentially tensor products of two Sturmian dynamical systems.