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# memo_has_moved_text();$C^*$-algebras of homoclinic and heteroclinic structure in expansive dynamics

Klaus Thomsen, IMF, Ny Munkegade, 8000 Aarhus C, Denmark

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: Volume 206, Number 970
ISBNs: 978-0-8218-4692-6 (print); 978-1-4704-0584-7 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00581-8
Published electronically: February 24, 2010
MathSciNet review: 2667385
MSC (2000): Primary 46L55; Secondary 37B10, 37D20, 46L05, 46L35, 46L80

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Chapters

• Chapter 1. The Ruelle algebra of a relatively expansive system
• Chapter 2. On the functoriality of the Ruelle algebra
• Chapter 3. The homoclinic algebra of expansive actions
• Chapter 4. The heteroclinic algebra
• Chapter 5. One-dimensional generalized solenoids
• Chapter 6. The heteroclinic algebra of a group automorphism
• Chapter 7. 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

### Abstract

We unify 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.