Synchronizing Dynamical Systems: Finitely presented systems and Ruelle algebras

TL;DR

This paper investigates the structure of $C^*$-algebras linked to finitely presented and general synchronizing systems, establishing their relation to Smale spaces and exploring properties like Poincaré duality in specific cases.

Contribution

It develops the theory of Ruelle algebras for synchronizing systems and relates them to finitely presented systems and Smale spaces, including examples where duality fails.

Findings

Ruelle algebras for finitely presented systems are explicitly characterized.

Ruelle algebras for a general synchronizing system are developed and related to Smale spaces.

An example of a sofic shift where Poincaré duality does not hold is provided.

Abstract

The main goals of the present paper are to determine the structure of the $C^\ast$-algebras associated to a finitely presented system and to develop the basic theory of the Ruelle algebras associated to a general synchronizing system. The later is related to the former in the sense that we show that Ruelle algebras associated to a finitely presented system are explicitly related to the Smale space case. Nevertheless, we give an example of a sofic shift where the Ruelle algebras are not Poincare dual (whereas this duality holds in the Smale space case). The relevant $C^\ast$-algebras are the synchronizing heteroclinic algebras that were introduced in our previous work on synchronizing systems. They are very much related to previous work of Thomsen, who in turn was building on work of Ruelle, Putnam, and Spielberg.