Poincaré duality and spectral triples for hyperbolic dynamical systems
Date
2010-07-15T23:01:51Z
Authors
Whittaker, Michael Fredrick
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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.
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Mathematics, C*-algebras, operator algebras, Smale spaces