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Analysis SeminarExport this event to calendar

Friday, September 15, 2017 — 3:30 PM EDT

Sascha Troscheit, Department of Pure Mathematics, University of Waterloo

"Random fractals, their dimensions and measures"

The Cantor set can be thought of as the unique compact set invariant under the maps 1/3x and 1/3x+2/3. In fact, any finite collection of contractions on a 'nice' space, say finite Euclidean space, has an unique associated compact invariant set. This collection of maps is called an iterated function system (IFS) as its invariant set can also be obtained by choosing any compact "seed set" and successively applying the IFS to this set. After infinitely many iterations, the obtained limit set is the invariant set.

This construction can be modified by randomly choosing different IFSs at different stages and we will explain several different ways of doing so. These models will expand on classical models such as Mandelbrot percolation and culminate with "Random Graph Directed Systems". 

We will focus mostly on dimension theoretic results of these sets, but -- given time -- will mention some measure theoretic results and more tenuous links to number theory. The main motivation will be a random analogue of the "implicit theorems" due to Kenneth Falconer that has so far only been conjectured.

MC 5417

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