![]() Nonstandard development of the Hausdorff metric - III. Addressed to all readers with an interest in fractals, hyperspaces, fixed-point theory, tilings and nonstandard analysis, this book presents its subject in. Nonstandard development of the vietoris topology - II. Includes bibliographical references (pages 160-161) and index These ideas have considerable scope for further development, and a list of problems and lines of research is included. ![]() This leads to new, elegant concepts (defined purely topologically) of self-similarity and fractality: in particular, the author shows that many invariant sets are "visually fractal", i.e. The last and most original part of the book introduces the notion of a "view" as part of a framework for studying the structure of sets within a given space. Hutchinson's invariant sets (sets composed of smaller images of themselves) is developed, with a study of when such a set is tiled by its images and a classification of many invariant sets as either regular or residual. A major feature is that nonstandard analysis is used to obtain new proofs of some known results much more slickly than before. The first part of the book develops certain hyperspace theory concerning the Hausdorff metric and the Vietoris topology, as a foundation for what follows on self-similarity and fractality. A tutorial, Princeton University Press, Princeton, NJ, 2006.Addressed to all readers with an interest in fractals, hyperspaces, fixed-point theory, tilings and nonstandard analysis, this book presents its subject in an original and accessible way complete with many figures. Strichartz R.S., Differential equations on fractals. Strichartz R.S., Fractals in the large, Canad. J. Renault J., Cartan subalgebras in $C^*$-algebras, Irish Math. Renault J., Cuntz-like algebras, in Operator Theoretical Methods (Timişoara, 1998), Theta Found., Bucharest, 2000, 371-386, math.OA/9905185. Theory of $C^$-algebras, Lecture Notes in Mathematics, Vol. Soc., Providence, RI, 2001, 35-46.īarnsley M.F., Fractals everywhere, 2nd ed., Academic Press Professional, Boston, MA, 1993.īlackadar B., Operator algebras. Fractals and hyperspaces Item Preview remove-circle Share or Embed This Item. France 125 (1997), 199-225.Īnantharaman-Delaroche C., Renault J., Amenable groupoids, in Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Contemp. pdf (367 kb) tex (22 kb)Īnantharaman-Delaroche C., Purely infinite $C^*$-algebras arising from dynamical systems, Bull. Renault-Deaconu groupoids fractafolds iterated function systems. We also show that when the bundle is locally compact the associated $C^*$-algebra is primitive and has a densely defined lower-semicontinuous trace. ![]() We prove that there is a natural action of a Renault-Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault-Deaconu groupoid itself. We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. Received February 04, 2014, in final form JPublished online June 28, 2014 Rieffel Groupoid Actions on FractafoldsĪ) Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402-5002, USAī) Department of Mathematics, University of Nevada, Reno, NV, 89557, USA SIGMA 10 (2014), 068, 14 pages arXiv:1311.3880 Ĭontribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Groupoid Actions on Fractafolds Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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