Maik Gröger

Abstract

The talk will begin by reviewing both classical and more recent results on the fractal dimensions of the well-known Weierstrass functions. We will then explore the Brownian continuum tree, focusing in particular on how it can be constructed via a change of metric applied to an excursion function on the unit interval. This approach extends to all excursion functions (or continuous circle maps), allowing us to associate a real (rooted) tree to each such function. In this broader context, I will discuss a 2008 result by Picard, which shows that the dimension theory of these trees is intimately linked to their contour function: specifically, the upper box dimension of a real tree coincides with the variation index of its excursion function. As the talk progresses, I will present further results that deepen this connection, revealing additional relationships between the fractal dimensions of the tree and those of the graph of its excursion function.