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$$C^*$$-Algebras of Homoclinic and Heteroclinic Structure in Expansive Dynamics
Klaus Thomsen, IMF, Aarhus, Denmark
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Memoirs of the American Mathematical Society
2010; 122 pp; softcover
Volume: 206
ISBN-10: 0-8218-4692-2
ISBN-13: 978-0-8218-4692-6
List Price: US$69 Individual Members: US$41.40
Institutional Members: US\$55.20
Order Code: MEMO/206/970

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.

• The Ruelle algebra of a relatively expansive system
• On the functoriality of the Ruelle algebra
• The homoclinic algebra of expansive actions
• The heteroclinic algebra
• One-dimensional generalized solenoids
• The heteroclinic algebra of a group automorphism
• A dimension group for certain countable state Markov shifts
• Appendix A. Étale equivalence relations from abelian $$C^*$$-subalgebras with the extension property
• Appendix B. On certain crossed product $$C^*$$-algebras
• Appendix C. On an example of Bratteli, Jorgensen, Kim and Roush
• Bibliography