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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

Étale equivalence relations and C*-algebras for iterated function systems

Korfanty, Emily Rose 22 December 2020 (has links)
There is a long history of interesting connections between topological dynamical systems and C*-algebras. Iterated function systems are an important topic in dynamics, but the diversity of these systems makes it challenging to develop an associated class of C*-algebras. Kajiwara and Watatani were the first to construct a C*-algebra from an iterated function system. They used an algebraic approach involving Cuntz-Pimsner algebras; however, when investigating properties such as ideal structure, they needed to assume that the functions in the system are the inverse branches of a continuous map. This excludes many famous examples, such as the standard functions used to construct the Siérpinski Gasket. In this thesis, we provide a construction of an inductive limit of étale equivalence relations for a broad class of affine iterated function systems, including the Siérpinski Gasket and its relatives, and consider the associated C*-algebras. This approach provides a more dynamical perspective, leading to interesting results that emphasize how properties of the dynamics appear in the C*-algebras. In particular, we show that the C*-algebras are isomorphic for conjugate systems, and find ideals related to the open set condition. In the case of the Siérpinski Gasket, we find explicit isomorphisms to subalgebras of the continuous functions from the attractor to a matrix algebra. Finally, we consider the K-theory of the inductive limit of these algebras. / Graduate

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