The product structure of finitely presented dynamical systems (Q1182825)

Finitely presented systems which were introduced by \textit{D. Fried} with the intention of generalizing symbolic description of dynamical systems have recently become of interest as a class of dynamical systems to which the extensive theory of Axiom A systems and sofic systems can be extended without incurring too many casualities. However, one property that gets lost in generalizing is the local product structure or canonical coordinates, which characterize strongly hyperbolic systems: with it the shadowing property also goes, although, as Fried shows, there are still finite Markov partitions. On the other hand, much of the theory on Axiom A systems can indeed also be formulated for finitely presented systems. As an example we can point out Baladi's paper, which gives a good account of how the theory on Gibbs and equilibrium states can be carried over to finitely presented systems. The questions treated here arise in a natural way from Markov partitions on Axiom A systems, for it is well known that an Axiom A diffeomorphism on some manifold \(M\) is semiconjugate to the shift on a subshift of finite type constructed by partitioning \(M\) in a certain way. However, as a subshift of finite type can be isomorphic only to an Axiom A diffeomorphism over a non-wandering set of zero dimension, the ``boundary set'', that is the set of points whose preimages in the shift consist of more than one point, contains essential information about the topological structure of the non-wandering set, despite the fact that it has measure zero for any ergodic measure which is positive on open sets. The present note resolves the question of canonical coordinates for finitely presented systems from the purely symbolic point of view, by establishing a combinatorial criterion as a necessary and sufficient condition for the existence of a local product structure.