The spacetime of a shift endomorphism
  The automorphism group of a one dimensional shift space over a finite alphabet exhibits different types of behavior: for a large class with positive entropy, it contains a rich collection of subgroups, while for many shifts of zero entropy, there are strong constraints on the automorphism group. We view this from a different perspective, considering a single automorphism (and sometimes endomorphism) and studying the naturally associated two dimensional shift system. In particular, we describe the relation between nonexpansive subspaces in this two dimensional system and dynamical properties of an automorphism of the shift. 
Van Cyr, John Franks, Bryna Kra
math.DS