Seminar 11.21.13

A Z^d shift of finite type is the set of all d-dimensional arrays of symbols from a finite set which avoid a finite set of ‘forbidden’ patterns; for example, the set of all ways of assigning a 0 or 1 to every site in Z^2 so that no two 1s are horizontally or vertically adjacent is a Z^2 shift of finite type. Any shift of finite type can be considered as a topological dynamical system, where the dynamics comes from the Z^d-action of ‘shifting’ by vectors in Z^d. A topological dynamical system was defined by Blanchard to have topologically completely positive entropy (or t.c.p.e.) if all of its nontrivial factors have positive topological entropy. (Here, ‘nontrivial’ means not consisting of a single fixed point) In some sense, t.c.p.e. means that there is no ‘hidden’ degenerate behavior within the dynamical system. Though t.c.p.e. is not easy to characterize in general, I will present a surprisingly simple characterization of t.c.p.e. for shifts of finite type.