Title

Equilibrium states and Cuntz-Pimsner algebras on Mauldin-Williams graphs

Year

2019

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

In 1980, Hutchinson introduced iterated function systems as mathematical models for fractals. Kajiwara and Watatani associated Cuntz-Pimsner algebras to iterated function systems in 2006, and the KMS states for these Cuntz- Pimsner algebras, with respect to the lifted gauge action, were characterised by Izumi, Kajiwara and Watatani in 2007 under some mild assumptions. The notion of an iterated function systems was generalised by Mauldin and Williams in 1988 to what we now know as Mauldin-Williams graphs, and in 2008, Ionescu and Watatani associated a Cuntz-Pimsner algebra to each Mauldin-Williams graph via a construction analogous to Kajiwara and Watatani's construction for iterated function systems. In this thesis, we characterise the KMS-state structure for the lifted gauge action on these Cuntz- Pimsner algebras, extending the analysis of Izumi, Kajiwara and Watatani on iterated function systems. The class of iterated function systems for which our analysis holds is strictly larger than those considered by Izumi, Kajiwara and Watatani. We obtain a characterisation of the KMS states of Ionescu and Watatani's Cuntz-Pimsner algebra associated to Mauldin-Williams graphs satisfying either the finite branch condition with escape (this includes the class of iterated function systems considered by Izumi, Kajiwara and Watatani), or the open set condition.

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Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.