# \(C^*\)-Algebras of Homoclinic and Heteroclinic Structure in Expansive Dynamics

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*Klaus Thomsen*

The author unifies various constructions of \(C^*\)-algebras from dynamical systems, specifically, the dimension group construction of Krieger for shift spaces, the corresponding constructions of Wagoner and Boyle, Fiebig and Fiebig for countable state Markov shifts and one-sided shift spaces, respectively, and the constructions of Ruelle and Putnam for Smale spaces. The general setup is used to analyze the structure of the \(C^*\)-algebras arising from the homoclinic and heteroclinic equivalence relations in expansive dynamical systems, in particular, expansive group endomorphisms and automorphisms and generalized 1-solenoids. For these dynamical systems it is shown that the \(C^*\)-algebras are inductive limits of homogeneous or sub-homogeneous algebras with one-dimensional spectra.

#### Table of Contents

# Table of Contents

## $C^{*}$-Algebras of Homoclinic and Heteroclinic Structure in Expansive Dynamics

- Preface vii8 free
- Chapter 1. The Ruelle algebra of a relatively expansive system 112 free
- Chapter 2. On the functoriality of the Ruelle algebra 1324
- Chapter 3. The homoclinic algebra of expansive actions 2334
- Chapter 4. The heteroclinic algebra 4152
- Chapter 5. One-dimensional generalized solenoids 5768
- Chapter 6. The heteroclinic algebra of a group automorphism 6778
- Chapter 7. A dimension group for certain countable state Markov shifts 8394
- Appendix A. Étale equivalence relations from abelian C*-subalgebras with the extension property 103114
- Appendix B. On certain crossed product C*-algebras 109120
- Appendix C. On an example of Bratteli, Jorgensen, Kim and Roush 115126
- Bibliography 119130