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Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

Thursday, July 15, 17:45 ~ 18:15 UTC-3

Intermediate Assouad-like dimensions

Ignacio García

CEMIM - IFIMAR, Universidad Nacional de Mar del Plata and CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Many notions of dimension have been introduced to help understand the geometry of (often `small') subsets of metric spaces, such as subsets of $\mathbb{R}^{n}$ of Lebesgue measure zero. Hausdorff and box dimensions are well known examples of such notions, and possibly less known are the upper and lower Assouad dimensions of a set $E$, which quantify the `thickest' or `thinnest' part of the space.

In the talk I will review some results related to a general class of `intermediate dimensions', referred to as the upper and lower $\Phi $-dimensions, which can roughly be thought of as local refinements of the box-counting dimensions where one takes the most extreme local behaviour, at scales `ruled' by a function $\Phi$. These $\Phi$-dimensions provide a range of bi-Lipschitz invariant dimensions between the box and Assouad dimensions. As the box and Assouad dimensions for a given set can all be different, the intermediate $\Phi$-dimensions provide more refined information about the local geometry of the set, such as detailed information about the scales at which one can observe extreme local behaviour.

Joint work with Kathryn Hare (University of Waterloo, Canada) and Kranklin Mendivil (Acadia University, Canada).

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