Dynamical properties of commuting automorphisms 1 Graz Summer School 2007 Tom Ward 1 These notes are an expanded account of lectures for the Summer School on Dynamical Systems and Number Theory, Graz, July 9-13, 2007. They describe some of the theory developed by Einsiedler, Kitchens, Lind, Miles, Schmidt and others of dynamical properties of group automorphisms. I am grateful to Richard Miles for comments. One specific warning: the footnotes are not intended to make sense until after all the lectures – and the lectures of the other speakers – have happened. The final 7 lectures given will cover 1–6 and one further topic. Draft dated November 16, 2009