Poincaré duality and spectral triples for hyperbolic dynamical systems
| dc.contributor.author | Whittaker, Michael Fredrick | |
| dc.contributor.supervisor | Putnam, Ian Fraser | |
| dc.date.accessioned | 2010-07-15T23:01:51Z | |
| dc.date.available | 2010-07-15T23:01:51Z | |
| dc.date.copyright | 2010 | en |
| dc.date.issued | 2010-07-15T23:01:51Z | |
| dc.degree.department | Department of Mathematics and Statistics | |
| dc.degree.level | Doctor of Philosophy Ph.D. | en |
| dc.description.abstract | We study aspects of noncommutative geometry on hyperbolic dynamical systems known as Smale spaces. In particular, there are two C*-algebras, defined on the stable and unstable groupoids arising from the hyperbolic dynamics. These give rise to two additional crossed product C*-algebras known as the stable and unstable Ruelle algebras. We show that the Ruelle algebras exhibit noncommutative Poincaré duality. As a consequence we obtain isomorphisms between the K-theory and K-homology groups of the stable and unstable Ruelle algebras. A second result defines spectral triples on these C*-algebras and we show that the spectral dimension recovers the topological entropy of the Smale space itself. Finally we define a natural Fredholm module on the Ruelle algebras in the special case that the Smale space is a shift of finite type. Using unitary operators arising from the Pimsner-Voiculescu sequence we compute the index pairing with our Fredholm module for specific examples. | en |
| dc.identifier.uri | http://hdl.handle.net/1828/2897 | |
| dc.language | English | eng |
| dc.language.iso | en | en |
| dc.rights | Available to the World Wide Web | en |
| dc.subject | Mathematics | en |
| dc.subject | C*-algebras | en |
| dc.subject | operator algebras | en |
| dc.subject | Smale spaces | en |
| dc.subject.lcsh | UVic Subject Index::Sciences and Engineering::Mathematics::Pure mathematics | en |
| dc.title | Poincaré duality and spectral triples for hyperbolic dynamical systems | en |
| dc.type | Thesis | en |