The largest set defined by an iterated function system
Speaker: Professor Micheal Barnsley (Canberra)
Time: 3:00PM
Date: Tuesday 1st April 2014
Location: Mathematics Seminar Room, Room AG 1.01, First Floor, Agriculture Building, UCD Belfield
Abstract:
In this lecture I will describe recent, to me very exciting, work concerning natural extensions of attractors of iterated functionsystems. The results so far are largely joint work with AndrewVince and Krystoph Lesniak. My goal will be to explain the newideas, the motivation for them, and the shape of the final "bigpicture". But next, here, I quote the abstract of one of a numberof recent papers on the topic, to give some flavour. (The lectureitself will be less technical.) We define and exemplify thecontinuations and the fast basin of an attractor of an IFS. Thenwe extend the standard symbolic IFS theory, concerning the dynamicsof a contractive IFS on its attractor, to a symbolic descriptionof the dynamics of a invertible IFS on a set that contains the fastbasin of a point-fibred attractor. We use this description to definethe fractal manifold, a new topological invariant, associated witha point-fibred attractor of an IFS. We establish relationshipsbetween the fractal manifold, the fast basin, and the set ofcontinuations of an attractor of an IFS. We establish how sectionsof projections, from code space to an attractor, can be extendedto yield sections of projections from code space to the f-manifoldand to the basin. We use these sections to construct transformationsbetween fractal manifolds and show how their projections extendfractal transformations between attractors to correspondingtransformations between fast basins of attractors. We presentpractical conditions that control the topological properties, suchas continuity, of these transformations. We show how a section ofa projection from code space to an attractor yields a unique addressfor each point on the fractal manifold, and how the set of addressesprovides a tiling of the manifold that project to diverse tilingsof the basin. Many standard tilings, and schemes for describingthem, as well as an abundance of new tilings, result from this newunified approach to fractal tiling. This work has implications tocoding theory and numeration systems.
Series: Analysis Seminar Series
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