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Synchronizing Dynamical Systems: Finitely presented systems and Ruelle algebras

Robin J. Deeley, Andrew M. Stocker

TL;DR

The paper extends the Ruelle algebra framework from Smale spaces to finitely presented and synchronizing systems by constructing synchronizing heteroclinic and Ruelle algebras through groupoid C*-algebras. It proves a central theorem linking the groupoids of a finitely presented system to those of its Smale-space covers via minimal $s$- and $u$-resolving extensions, enabling the transfer of K-theory and structural results. Consequently, many Smale-space C*-algebraic properties—such as Morita equivalences, nuclear dimension, and real rank zero—carry over to the finitely presented setting, under suitable hypotheses, and the corresponding crossed products behave coherently. A counterexample with a sofic shift demonstrates that Poincaré duality can fail in this broader context, underscoring fundamental differences between finitely presented and Smale-space dynamics. Overall, the work clarifies how finitely presented and synchronizing systems inherit and diverge from Smale-space phenomena in the realm of groupoid and C*-algebraic structures.

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.

Synchronizing Dynamical Systems: Finitely presented systems and Ruelle algebras

TL;DR

The paper extends the Ruelle algebra framework from Smale spaces to finitely presented and synchronizing systems by constructing synchronizing heteroclinic and Ruelle algebras through groupoid C*-algebras. It proves a central theorem linking the groupoids of a finitely presented system to those of its Smale-space covers via minimal - and -resolving extensions, enabling the transfer of K-theory and structural results. Consequently, many Smale-space C*-algebraic properties—such as Morita equivalences, nuclear dimension, and real rank zero—carry over to the finitely presented setting, under suitable hypotheses, and the corresponding crossed products behave coherently. A counterexample with a sofic shift demonstrates that Poincaré duality can fail in this broader context, underscoring fundamental differences between finitely presented and Smale-space dynamics. Overall, the work clarifies how finitely presented and synchronizing systems inherit and diverge from Smale-space phenomena in the realm of groupoid and C*-algebraic structures.

Abstract

The main goals of the present paper are to determine the structure of the -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 -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.
Paper Structure (14 sections, 25 theorems, 93 equations, 2 figures)

This paper contains 14 sections, 25 theorems, 93 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Expansive dynamical systems
  3. Local Product Structure
  4. Metrics on expansive systems
  5. Maps
  6. Groupoids
  7. The Local Conjugacy Relation
  8. The Stable/Unstable Local Conjugacy Relations
  9. $C^\ast$-Algebras from Expansive Dynamical Systems
  10. The Ruelle algebras
  11. Main results
  12. Groupoid results
  13. $C^\ast$-algebraic results
  14. Counterexample to Poincare duality

Key Result

Theorem 2

Suppose that $(X, \varphi)$ is a mixing Smale space and $S\rtimes {\mathbb Z}$ and $U\rtimes {\mathbb Z}$ denote its stable Ruelle and unstable Ruelle algebras respectively. Then $S\rtimes {\mathbb Z}$ and $U\rtimes {\mathbb Z}$ are Poincare dual and

Figures (2)

  • Figure 1: The left Fischer cover of the irreducible sofic shift $X$ discussed in Section \ref{['sec:counterexample']}.
  • Figure 2: The right Fischer cover of the irreducible sofic shift $X$ discussed in Section \ref{['sec:counterexample']}.

Theorems & Definitions (64)

  • Conjecture 1
  • Theorem 2
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Lemma 1.7
  • Definition 1.8
  • ...and 54 more