Mixing Shifts of Finite Type with Non-Elementary Surjective Dimension Representations

The dimension representation has been a useful tool in studying the mysterious automorphism group of a shift of finite type, the classification of shifts of finite type, and surrounding problems. However, there has been little progress in understanding the image of the dimension representation. We discuss the importance of understanding the image of the dimension representation and discuss the candidate range for the fundamental case of mixing shifts of finite type. We present the first class of examples of mixing shifts of finite type for which the dimension representation is surjective necessarily using non-elementary conjugacies. 1 Statement of Results The group of automorphisms of a shift of finite type (X ,σX), denoted Aut(σX), has been a vital tool in attempts to classify shifts of finite type, but in general, Aut(σX) is large and mysterious. For example, the automorphism group of the (2-sided) 2-shift is countably infinite, is residually finite, is not finitely generated, and contains a copy of many groups including every finite group, the free group on infinitely many generators, but not any group with unsolvable word problem [2]. The dimension group of a shift of finite type is a more tractable group to study. In fact, the dimension group is the direct limit group of the action on Zn by a n by n non-negative integral matrix presentation. Let A be a square non-negative integral matrix presenting the shift of finite type (XA,σA), and let Aut(Â) denote