Ergodic Theory and Dynamical Systems

Invariants for subshifts via nested sequences of shifts of finite type

a1 Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany (e-mail:


To obtain connections between grammars of languages and subshifts, Krieger introduced the idea of associating to a subshift X a certain increasing sequence of shifts of finite type X_{n} sitting inside X. Their union is the set of presynchronizing points. A system is called presynchronized if there is a sequence of irreducible components C_{n} \subset X_{n}, n \in \mathbb{N}, increasing with n and with dense union (a characterization of coded systems shows that presynchronized systems are coded). We investigate the uniqueness (up to eventual equality) of such sequences. For that an equivalence relation on the periodic presynchronizing points captures all the possible choices of increasing sequences of irreducible components. Dense equivalence classes correspond to dense unions. There is a one-sided and a two-sided theory. In the one-sided setting we show the uniqueness of a dense equivalence class and identify the right presynchronized systems as the well-known half-synchronized systems. The two-sided theory turns out to be more complicated (and interesting). There can be more than one dense class. Many examples are discussed in detail. There is a natural ordering on the set of (dense) equivalence classes. For each finite-order structure we find a representative in the countable class of systems which are intersections of Dyck shifts with shifts of finite type. Finally we discuss the relation to other known classes of subshifts, which is more subtle than in the one-sided setting. Almost all systems considered in this work will be coded.

(Received November 17 1999)
(Revised March 29 2000)