Minimising Hausdorff dimension under Hölder equivalence
Analysis and Geometry Seminar
21st February 2019, 3:00 pm – 4:00 pm
Howard House, 4th Floor Seminar Room
Conformal dimension is a powerful tool in the study of metric spaces. It was originally introduced to study rank one symmetric spaces, and is now used to study boundaries of hyperbolic groups and other fractal metric spaces. I will introduce the “Hölder dimension” of a metric space, a Hölder-equivalence parallel of conformal dimension, and relate it to other notions of dimension, such as topological, capacity, and Hausdorff dimension. For example, for self-similar spaces, Hölder dimension is equal to topological dimension. In the process, I’ll illustrate that any compact, doubling metric space can be mapped into Hilbert space, bi-Hölder onto its image, with good control on the Hausdorff dimension of the image.