Title: Hausdorff measure, fractals, and non-integer dimensions

 

Abstract: The Hausdorff measure provides a general notion of size for “low-dimensional” sets in a metric space, which is particularly useful when the traditional notions, such as the Lebesgue measure, fail to provide meaningful information. On the other hand, it is nontrivial to rigorously define the length or area for objects embedded in a high-dimensional space and even the notion of dimension. For example, a 1×1 solid square has an infinite length, an area of 1 square unit, and zero volume. In this example, there are three notions of dimension, but the two-dimensional notion seems “just right” for measuring the size of a solid square. In this talk, we will begin with constructing the Hausdorff measure and highlighting some of its key properties as well as its relationship with the Lebesgue measure. Next, we will explore how properties of the Hausdorff measure are used to define a new notion of dimension, the Hausdorff dimension, which can take non-integer values and is also “just right” for the study of fractals. Finally, we will discuss a technique to compute the Hausdorff dimension of strictly self-similar sets such as the middle third Cantor set.