Dynamical covering arguments via large deviations and non-convex optimization
Alex Rutar (St Andrews)
Most classical notions of fractal dimensions (such as the Hausdorff, box, and Assouad dimensions) are defined in terms of optimal covers, or families of balls minimizing some form of cost function of their radii. For general sets, the optimal covers can be forced to essentially have arbitrary complexity. But for sets satisfying some form of dynamical invariance (which is the case for the majority of well-studied ‘fractal’ sets), one hopes that the underlying dynamics can be used to inform the optimal choice of cover in a meaningful way. In this talk, I will present some techniques drawing on insights from large deviations theory and continuous optimization theory which have proven to be useful technical tools in dimension theory. If time permits, I will discuss how these techniques, generalize to the dimension theory and multifractal analysis of sets invariant under certain families of affine transformations in the plane (dimension theory joint with Amlan Banaji, Jonathan Fraser, and István Kolossváry; multifractal analysis with Thomas Jordan and István Kolossváry).
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